The representation theory of the increasing monoid
Sema Güntürkün Andrew Snowden



We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.


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§ INTRODUCTION


The increasing monoid  is the monoid of all (tame[See <ref> for the definition of tame.]) order-preserving injections from the set {1,2,…} of positive integers to itself. The purpose of this paper is to investigate the representation theory of . Our results, outlined in <ref> below, give a fairly comprehensive picture of the structure of representations.



 §.§ Motivation


We begin by explaining why the representation theory of  is worthy of study, and how it connects to some other topics.



  §.§.§ Symmetric ideals


Cohen <cit.> proved that ideals in the infinite variable polynomial ring R=[x_1,x_2,…] that are stable under the infinite symmetric group _∞ satisfy the ascending chain condition—that is, R is _∞-noetherian—and used this theorem to prove a finiteness property of the variety of metabelian groups. This theorem has received intense interest in the last decade (see, e.g., <cit.>) as researchers from disparate areas have realized its utility in establishing stabilization phenomena; see <cit.> for a good introduction. In all treatments of _∞-ideals, the increasing monoid has played a key role: it respects the natural monomial order on R, and thus allows one to carry out equivariant Gröbner theory, which is crucial for establishing finiteness results. The theory developed in this paper should therefore shed some new light on the theory of _∞-ideals. In fact, we have already used the results of this paper to prove a new result in this direction <cit.>.



  §.§.§ -modules


A related topic, which has also seen a surge of interest in recent years, is the representation theory of combinatorial categories, especially the categories  and ; see, for example, <cit.>. The category of -modules is (essentially) equivalent to the category of graded -modules studied in this paper (see <ref>). Thus the results of this paper establish a structure theory for -modules. In fact, this paper can be viewed as an “ analog” of the paper <cit.>, which establishes structural results for -modules. As a consequence, the results herein should be of use wherever -modules are employed. For example, recent work of Putman, Sam, and the second author <cit.> shows that the homology of unipotent groups admits the structure of a finitely generated -module (in certain cases). It would be interesting to apply the results of this paper to refine the results of loc. cit.; see <ref> for more details.



  §.§.§ Semi-simplicial vector spaces


The celebrated Dold–Kan theorem shows that the category of simplicial vector spaces is equivalent to the category of chain complexes of vector spaces supported in non-negative degrees. Chain complexes of vector spaces have a simple structure: there are two indecomposable objects up to shift. Thus, via Dold–Kan, simplicial vector spaces are also rather simple, and one can establish a complete structure theory for them.

Semi-simplicial vector spaces, on the other hand, are far more complicated. However, the category of semi-simplicial vector spaces is (essentially) equivalent to the category of graded -modules (see <ref>). Thus the results of this paper can be seen as a structure theory for semi-simplicial vector spaces.



  §.§.§ Commutative algebra


There are two interesting connections between commutative algebra and the theory of -modules. First, many definitions, constructions, and results for -modules are direct analogues of concepts in commutative algebra. For example, we define versions of Hilbert series, Krull dimension, Tor, Betti tables, the Koszul complex, and local cohomology for -modules, and establish familiar relationships such as that between Krull dimension and the degree of the pole of the Hilbert series at t=1. The richness of commutative algebra therefore hints that there may be considerable depth to the theory of -modules. In this direction, we formulate a version of the Boij–Söderberg problem for -modules (<ref>).

Second, a direct overlap between commutative algebra and -modules is found in monomial ideals. Precisely, we introduce an -module ^n, called the nth principal module, which is isomorphic, as a vector space, to the n-variable polynomial ring. We show that, under this linear isomorphism, monomial submodules of ^n correspond bijectively to monomial ideals of the polynomial ring (Proposition <ref>.). An interesting problem is to understand how properties of monomial ideals relate to the corresponding properties of monomial -modules (<ref>).



  §.§.§ Shuffle algebras


An (n,m)-shuffle is a bijection [n] ⨿ [m] → [n+m] that is order-preserving on each factor. A shuffle algebra is a graded vector space A = ⊕_n ≥ 0 A_n equipped with a multiplication map ∗_σ A_n × A_m → A_n+m for each (n,m)-shuffle σ, satisfying some natural axioms (see <cit.>). Commutative shuffle algebras are the “ordered version” of twisted commutative algebras, which have played a prominent role in representation stability. We show that the category of graded -modules is (essentially) equivalent to the category of modules over the commutative shuffle algebra A freely generated by one element in degree one (<ref>). Thus the results of this paper can be viewed as a structure theory for A-modules. While shuffle algebras have cropped up in several places <cit.>, this paper represents the first effort to understand their modules in any detail (as far as we know). As this paper is entirely devoted to the module theory of A, which is the simplest shuffle algebra, there is likely much more to discover in this direction.



 §.§ Fundamental definitions
 

Before stating our results, we must first introduce a few key concepts. Fix a field . By an -module, we will mean a representation of  over , or, equivalently, a left module over the monoid algebra []. As general -modules are rather wild, we focus on two more reasonable classes:


  * Let _>n be the submonoid of  consisting of those elements that fix each of the numbers 1, …, n. We say that an -module M is smooth if every vector x ∈ M is fixed by some _>n, where n depends on x. We let () denote the category of smooth representations.

  * A graded -module is an -module M equipped with a grading M=⊕_n ≥ 0 M_n such that σ∈ maps M_n to M_σ(n), and acts by the identity on M_n if σ(n)=n. (We put σ(0)=0 for all σ∈.) We let () denote the category of graded -modules.

This paper is concerned with determining the structure of these two categories.

There are three families of -modules that will play a central role in our work:


  * For r ∈, let ^r be the one-dimensional graded -module that is concentrated in degree r, and where σ acts by the identity if σ(r)=r and by 0 otherwise. These are exactly the simple objects in the category ().

  * For r ∈, let ^r be the graded -module that has for a basis elements e_i_1,…,i_r where 1 ≤ i_1 < ⋯ < i_r are integers, and σ∈ acts in the obvious way, i.e., σ e_i_1,…,i_r=e_σ(i_1),…,σ(i_r). The element e_i_1,…,i_r is assigned degree i_r. We call ^r the rth principal module. The principal modules are exactly the indecomposable projective objects in ().

  * A constraint word is a word λ=λ_1 …λ_r in the alphabet {, }. We define a graded -module ^λ, called the standard module associated to λ, as follows. First, ^λ has for a basis elements of the form e_i_1,…,i_r where 1 ≤ i_1 < ⋯ <i_r, subject to the following constraint: if λ_k= then i_k-i_k-1=1 (with the convention i_0=0). We define σ e_i_1,…,i_r to be e_σ(i_1),…,σ(i_r) if this is a basis element of ^λ, and 0 otherwise. The ^λ interpolate between the ^r and the ^r: indeed, if λ=^r then ^λ=^r, while if λ=^r then ^λ=^r.

We let ^r, ^r, and ^λ be the modules obtained from ^r, ^r, and ^λ by forgetting the grading. These are all smooth. It is still true that the ^r are the simple objects of (), however, it is no longer true that the ^r are projective; in fact, () has no non-zero projective objects.



 §.§ Main results
 

We now describe some of our main results in the graded case. Many (but not all) of these results have analogs in the smooth case. We note that all results hold over any coefficient field .



  §.§.§ The Grothendieck group

The classes of the standard modules ^λ form a basis for the Grothendieck group () of the category ()^ of finitely generated graded -modules (Theorem <ref>). Moreover, () admits a (non-symmetric) monoidal operation ⊙, called the concatenation product, that endows () with the structure of a non-commutative ring. As such, it is isomorphic to the non-commutative polynomial {a,b} in two variables (Proposition <ref>). The isomorphism {a,b}→() takes a to [^1] and b to [^1], and a general word λ to [^λ]. We also determine the effective cone in () (<ref>).



  §.§.§ Generators for the derived category

The standard modules ^λ generate ^b_(()) as a triangulated category (Theorem <ref>). This implies that the [^λ] span (), but is far stronger (and more useful): for example, it typically allows one to prove finiteness statements about a derived functor by simply computing what the functor does to standard modules.



  §.§.§ Hilbert series

Let M be a finitely generated graded -module. The Hilbert series of M is defined to be

_M(t) = ∑_n ≥0 (M_n) t^n.

It was shown in <cit.>, using the language of -modules, that this is a rational function. We take this result a step further. We first observe that (M_n) can be viewed as the multiplicity of the simple object ^n in M. We extend this by defining an invariant _λ(M) that is, roughly speaking, the multiplicity of ^λ in M (<ref>). We then define the non-commutative Hilbert series of M to be

_M = ∑_λ _λ(M) λ,

where the sum is over all words λ in a and b. We regard this as a non-commutative power series in the variables a and b. We prove that it is rational (Theorem <ref>).



  §.§.§ Classification of injective modules

We explicitly construct the injective envelope ^λ of ^λ, and find that it is finitely generated. We show that every indecomposable injective object of () is isomorphic to ^λ for some λ. When λ=^r, we find that ^λ=^λ=^r. Thus the principal objects ^r are injective, and so all projective objects of () are injective. One interesting feature of ^λ is that it admits a standard filtration, that is, a finite length filtration where the graded pieces are standard modules. (See <ref> for these claims.) 



  §.§.§ Injective resolutions

Every finitely generated graded -module M has finite injective dimension (Theorem <ref>). In fact, M admits an injective resolution of the form M → I^∙ where each I^n is a finite sum of indecomposable injectives, and I^n=0 for n ≫ 0. As a consequence, we find that () admits a canonical pairing, namely the Euler characteristic of ^∙(-,-). We give a recurrence that allows one to effectively compute this pairing (<ref>).



  §.§.§ Projective resolutions

Let M be a finitely generated graded -module. Just as in commutative algebra, one can speak of the minimal projective resolution of M and its linear strands. We show that only finitely many linear strands are non-zero; in other words, the (Castelnuovo–Mumford) regularity of M is finite. Furthermore, we show that each linear strand (after some simple transformation) canonically admits the structure of a finitely generated graded -module. Thus the minimal resolution of M exhibits strong finiteness properties. We note, however, that if M is not projective then it has infinite projective dimension.

We define the graded Betti table β(M) of M in <ref> just as in commutative algebra: the ith row records the number of generators appearing in the ith linear strand of various degrees. The finiteness result for regularity ensures that β(M) has only finitely many non-zero rows. The finite generation result for each linear strand implies that for each i, the function j ↦β(M)_i,j is eventually a polynomial in j. (See Theorem <ref> for these statements.) An interesting open problem is to find an analog of Boij–Söderberg theory in this setting to describe the cone of Betti tables; see <ref>.



  §.§.§ Koszul duality

We construct a canonical equivalence ^b_(())^→^b_(()) that squares to the identity (Corollary <ref>). This can be seen as a version of Koszul duality. The existence of this functor on the full derived category (()) is fairly formal. The fact that it preserves the bounded finitely generated subcategory is a much deeper result; in fact, it is essentially a reformulation of the properties of projective resolutions mentioned above. The duality functor induces an involution of the ring () ≅{a,b} that takes a to -b and b to -a.



  §.§.§ Level categories

We define a notion of level for graded -modules. It turns out that the level of a module is equal to its Krull–Gabriel dimension, though that is a non-trivial result (Proposition <ref>). We let ()_≤ r be the full subcategory of () spanned by the objects of level ≤ r. These categories filter (); we refer to this as the level filtration. We completely determine the structure of the graded pieces ()_r=()_≤ r/()_≤ r-1. Precisely, we show that ()_r is equivalent to the category of multi-graded ^r+1-modules that are locally of finite length; equivalently, this can be described as the (r+1)st tensor power of the category ()^. (See Corollary <ref>.) We also show that ^b_(()) admits a semi-orthogonal decomposition where the pieces are {^b_(()_r)}_r ≥ 0 (Theorem <ref>).



 §.§ Open problems
 

While our work gives a fairly good picture of the structure of -modules, it also suggests a number of further avenues of inquiry. We highlight a few potentially interesting problems here.



  §.§.§ The Boij–Söderberg problem
 
As mentioned, we show that the Betti table β(M) of a finitely generated graded -module is reasonably well-behaved: it has finitely many non-zero rows and each row is eventually a polynomial function. It therefore seems reasonable to consider the Boij–Söderberg problem in this context: can one describe the cone of all Betti tables?

We have one interesting observation to offer regarding this problem. In classical Boij–Söderberg theory, the pure resolutions constructed by Eisenbud–Fløystad–Weyman <cit.> play an important role (in characteristic 0). We note that these complexes have manifestations in (): indeed, the EFW complex is built out of Schur functors, and thus can be transformed, via Schur–Weyl duality, to a complex of -modules; one can then restrict to  and apply the equivalence between -modules and (). This yields a large class of pure resolutions in ().



  §.§.§ Regularity bounds
 
For a graded -module M, let t_i(M) be the maximal non-zero degree occurring in the ith Tor group _i (M), as defined in <ref>. The regularity ρ(M) is defined as the maximum value of t_i(M)-i over i ≥ 0. Alternatively, ρ(M) is the index of the final non-zero row in the Betti table β(M). As mentioned, we show that ρ(M) is finite when M is finitely generated.

Analogous results are known in the setting of -modules and _d-modules. In fact, even stronger results are known there. Church and Ellenberg <cit.> showed that the regularity of an -module can be bounded in terms of just t_0(M) and t_1(M), while Sam and the second author <cit.> proved that the regularity of an _d-module can be bounded in terms of t_0(M), …, t_n(M) for n ≈14 d^2.

Given these results, it is natural to wonder if such an improvement can exist for graded -modules. We suspect that there may be such a result in the following form: given r ≥ 0, there exists n(r) such that ρ(M) can be bounded in terms of t_0(M), …, t_n(r)(M) for all M ∈()_≤ r. Recall that ()_≤ r consists of those modules of level (or, equivalently, Krull–Gabriel dimension) at most r. We have not seriously attempted to prove this statement, however.



  §.§.§ Other examples of -modules
 
We introduce many invariants of -modules, such as local cohomology, (non-commutative) Hilbert series, Betti tables, and so on. However, we have only computed these invariants on standard modules, and the results are typically far too simple to be representative of the general situation. It would therefore be interesting to compute these invariants on more interesting -modules. Here are a few potentially interesting sources of examples:


  * As mentioned, we establish a bijection between monomial submodules of ^r and monomial ideals in [x_1, …, x_r]. Thus each monomial ideal gives an example of an -module.

  * Suppose  has characteristic 0. Let L_λ be the -module that in degree n ≥|λ| is the irreducible corresponding to λ[n]=(n-|λ|, λ_1, λ_2, …), and that vanishes in lower degrees. We can restrict L_λ to  and then apply the equivalence with () to obtain a graded -module L_λ'.

  * Let (_∞) be the category of algebraic representations of the infinite symmetric group, as studied in <cit.>. There is a natural restriction functor (_∞) →(), and so every algebraic representation of _∞ yields a smooth -module.

A theorem of Hochster <cit.> relates the local cohomology of a monomial ideal (in the classical sense) to the topology of the associated Stanley–Reisner complex. A theorem of Sam and the second author <cit.> relates the local cohomology of L_λ (in the sense of -modules) to the Borel–Weil–Bott rule. It would be interesting to see how the local cohomology of the corresponding -modules (in the sense of this paper) compares in these cases.

















  §.§.§ Homology of unipotent groups
 
Let U_n(R) ⊂_n(R) be the group of strictly upper-triangular matrices, where R is a ring whose additive group is finitely generated. Let M_n=_n(U_n(R), ) be the group homology of U_n(R) with coefficients in the field . By <cit.>, the collection of groups {M_n}_n ≥ 0 admits the structure of a finitely generated -module; we can therefore regard it as a finitely generated graded -module. Can one describe this -module from the perspective of this paper? For example, can one give its class in ()? When R= and =, the group M_n is understood <cit.>, and so this could be a reasonably feasible problem. (We note that describing the class of M in () is strictly stronger than determining the dimensions of the M_n's, since the Hilbert series map () → t has a large kernel; in other words, the class in the Grothendieck group records some non-trivial information about the transition maps.)



  §.§.§ Some  calculations
 
As mentioned, () admits a canonical pairing, given by the Euler characteristic of ^∙, and we have given recurrences that allows one to effectively compute the pairing (<ref>). Can one give a closed-form formula for the pairing? Or, even better, can one compute the individual  groups between standard modules?



  §.§.§ _d-modules for d>1

The results of this paper can be viewed as a structure theory for -modules. A natural problem is to extend these results to _d-modules, for d>1. In other words, if this paper is the “-analog” of the paper <cit.> on -modules, we are asking for the “_d-analog” of the paper <cit.> on _d-modules. As <cit.> suggests, there should be some geometry present in the general theory of _d-modules that is not seen in the d=1 case, which means the results and methods of this paper will probably not trivially generalize to the d>1 case.


























 §.§ Technical highlights


Since this is a very long paper without a singular goal, it is difficult to summarize the main ideas of proofs in a useful way. Instead, we give a sampling here of some important constructions and intermediate results, which will hopefully provide the reader some direction.

There are three really important technical theorems we wish to highlight:


  * Theorem <ref> (the canonical grading): every finite length (or even locally finite length) smooth -module admits a canonical grading. In other words, the forgetful functor Φ()^→()^ is an equivalence of categories. This result surprised us when we discovered it, and is still somewhat surprising to us now. This theorem is needed to even define the completion functor Ξ (discussed below).

  * Theorem <ref> (the multiplicity one theorem): the trivial representation has multiplicity one in the principal module ^r. In other words, if K denotes the kernel of the augmentation map ^r → then no subquotient of K is isomorphic to the trivial representation. This theorem is the key ingredient used to establish the main properties of the truncation functors (discussed below), which have a number of uses.

  * Theorem <ref> (the theorem on level). We define the rank of a constraint word λ, or the associated standard module ^λ, to be the number of a's in λ. For example, ^r has rank r, while ^r has rank 0. We say that a finitely generated smooth -module has level ≤ r if it admits a finite length filtration such that each graded piece is isomorphic to a subquotient of some standard module of rank ≤ r. The theorem on level (which holds in both the smooth and graded cases) states that any proper quotient of a rank r standard module has level <r. It is an extremely important result on the structure of -modules. For example, it immediately implies that the standard modules generate the derived category, and thus span the Grothendieck group.


Most of the work in this paper is devoted to studying various constructions (or functors) involving -modules. There are four functors in particular that are extremely important:



  * The forgetful functor Φ() →(), which simply forgets the grading. It may seem like a simple operation, and it is, but it has some non-trivial properties: for example, it is continuous (Proposition <ref>, not too hard) and takes injective objects to injective objects (Corollary <ref>, quite hard). Essentially by definition, we have Φ(^λ)=^λ.


  * The direct limit functor Ψ() →(). Given a graded -module M, we can form the following directed system of vector spaces:

M_1 α_1⟶ M_2 α_2⟶ M_3 α_3⟶ ⋯
(Here α_i is one of the basic generators of , see <ref>.) We define Ψ(M) to be the direct limit. It naturally carries the structure of a smooth -module. We show that Ψ realizes () as the Serre quotient of () by a certain category of torsion modules (Proposition <ref>). The effect of Ψ on standard modules is easy to describe (Proposition <ref>): if λ ends in the letter a, that is, it has the form μ a, then Ψ(^λ)=^μ; otherwise Ψ(^λ)=0.


  * The invariants functor Γ() →(). Given a smooth -module M, we let Γ(M)_n denote the invariants of M under the monoid ^≥ n introduced in <ref>. It is not hard to see that Γ(M)=⊕_n ≥ 0Γ(M)_n admits the structure of a graded -module. We show that it is naturally the right adjoint of Ψ (Proposition <ref>). Of the main functors studied in this paper, Γ is the only one that is not exact. Using the completion functor (discussed below), we show that the derived functor of Γ has amenable finiteness properties (Propositions <ref> and <ref>). The effect of Γ on standard modules is quite simple: Γ(^λ)=^λ a.


  * The completion functor Ξ() →(), which is the most difficult of the four. Given a smooth -module M, the truncation τ^<n(M) (discussed below) is locally of finite length; therefore, it admits a canonical grading. We define Ξ(M) to be the inverse limit of the truncations in the graded category. We show that Ξ is exact and is naturally left adjoint to Φ. (This is why Φ takes injectives to injectives.)  The unit map M →Φ(Ξ(M)) is always injective, and the cokernel is in a sense smaller. This yields an exact sequence that leads to some important inductive arguments; for example, this is how we study the derived functor of Γ. With great effort, we compute the effect of Ξ on standard modules (Proposition <ref>): writing λ=μ a^n, where μ does not end in an a, we have Ξ(^λ)=⊕_0 ≤ i ≤ n^μ a^i.


There are a few other functors worth mentioning here:


  * The concatenation product ⊙ (<ref>). Given a graded -module M and a smooth (or graded) -module N, we define an interesting  action on the tensor product M ⊗ N. We denote the resulting smooth (or graded) -module by M ⊙ N. The effect on standard modules is what one might guess (Proposition <ref>): ^λ⊙^μ = ^λ⊙μ, where λ⊙μ denotes the concatenation of the two words. In particular, we see that we can build a general standard module ^λ by successively concatenating ^1 and ^1. This is a useful observation, as it sometimes allows us to prove results for standard modules just by proving them for simple and principal modules (this idea is used, in effect, in the proof of Theorem <ref>).


  * The transpose functor † (<ref>). Given a graded -module M, we define M^† to be the graded -module with the same underlying graded vector space, but where the action of  is “transposed:” on the nth graded piece, the actions of α_i and α_n+1-i are interchanged. The effect on standard modules is straightforward (Proposition <ref>): (^λ)^†=^λ^†, where λ^† denotes the reversed word. The transpose functor can be handy in constructing more complicated operations: for example, when acting on standard modules, Γ, Ψ, and Ξ all affect the rightmost letters; if one wanted the leftmost letters affected instead, simply throw in a transpose functor.


  * The truncation functors τ_≥ r and τ^<r (<ref>). Given a smooth -module, we show that there exists a unique exact sequence

0 →τ_≥r(M) →M →τ^<r(M) →0

such that τ_≥ r(M) has no subquotient isomorphic to ^s with s<r, and τ^<r(M) has no subquotient isomorphic to ^s with s ≥ r. These functors are exact: this is a non-trivial result (Proposition <ref>) that relies on the multiplicity one theorem. We have already seen that these functors figure into the very definition of the important completion functor. They also have other uses: for example, we use them to prove the the trivial module is injective (Corollary <ref>).


  * The shift functor Σ (<ref>) and the coinduction functor  (<ref>). We have an isomorphism of monoids i →_≥ 2 given by α_j ↦α_j+1; of course, _≥ 2 is also a submonoid of . The shift functor Σ is defined by restricting an -module to _≥ 2, and then transferring this back to an -module via i. Similary, the coinduction functor  is defined by first transferring the -module to an _≥ 2-module via i, and then applying coinduction along the inclusion. These functors are both exact, and (Σ, ) forms an adjoint pair (Proposition <ref>). From this, we see that coinduction takes injective objects to injective objects. For this reason, it plays an important role in our study of injectives.



As one can see, we develop quite an extensive toolkit. This greatly simplifies—and in some cases simply makes possible—the task of proving our main theorems. To see the machinery in action, take a look at the proofs of Theorem <ref> (the theorem on level) or Theorem <ref> (on the structure of the Grothendieck group), where many of these functors, and their basic properties, are employed in unison.



 §.§ Outline


We now outline the contents of the paper. In addition to summarizing the contents of each section, we hope that this provides a rough guide to the logical structure of our results (in contrast to the previous section, which paid no heed to logical order).

<ref>. The increasing monoid.
We establish some basic properties of the increasing monoid  and its generators {α_i}_i ≥ 1.

<ref>. Representation categories.
We introduce the various categories of representations and establish some technical categorical results. We also explain the connections to -modules, semi-simplicial vector spaces, and shuffle algebras.

<ref>. Monomial modules.
We introduce monomial modules generally, and the three monomial modules ^r, ^r, and ^λ specifically. We establish some basic properties of these modules (e.g., every smooth representation is a quotient of a sum of ^r's).

<ref>. Gröbner theory.
We develop two versions of Gröbner theory, and use them to establish some facts about representations. The first version studies submodules of ^r by relating them to monomial submodules. The main application of this version is the theorem that finitely generated -modules (either smooth or graded) are noetherian. This result was essentially already known, but the smooth case does not seem to appear in the literature, so we have included a proof. The second version of Gröbner theory studies inhomogeneous submodules of an arbitrary graded module by relating them to homogeneous submodules. We use this to relate certain properties of graded modules and their underlying smooth module (such as indecomposability).

<ref>. Concatenation, shift, and transpose.
We introduce these three fundamental operations, study how they interact with each other, and compute their effect on standard modules.

<ref>. Finite length modules.
We show that the ^r's and ^r's account for all simple modules. We then compute the ^1 groups between them, by directly analyzing the 2 × 2 matrices that define an extension. A consequence of these computations is that the trivial representation is injective and projective in the category ()^ of smooth modules that are locally of finite length. A second consequence is the existence of the canonical grading (every smooth -module that is locally of finite length admits a canonical grading).

<ref>. Multiplicities.
For n ∈, we let μ_n(M) be the multiplicity of the simple object ^n in the smooth -module M. We prove the important multiplicity one theorem, which states μ_0(^r)=1. The multiplicity one theorem has a number of consequences, e.g., we deduce from it that μ_n(M) is finite for all n whenever M is finitely generated. We also introduce the truncation functors τ_≥ r and τ^<r. Using the multiplicity one theorem, we show that these functors are exact. From this, we deduce that injective objects of ()^ remain injective in (). In particular, we find that the trivial representation is injective in ().

<ref>. () as a Serre quotient of ().
We introduce the Ψ and Γ functors, and establish their main properties: namely, that they form an adjoint pair and realize () as a Serre quotient of (). We also compute their effect on standard modules. Finally, we show that any smooth module that admits a grading (such as ^λ) is Γ-acyclic. This result will be important later when we study the derived functors of Γ.

<ref>. Completions.
We define the completion functor Ξ, and show that it is exact and naturally the left adjoint of Φ. We note that the definition of Ξ hinges on the existence of the canonical grading and the truncation functors. We explicitly compute Ξ on standard modules. Finally, we give a number of applications of the completion functor, such as: (a) injectivity of principle modules; (b) finiteness properties of Γ; and (c) finiteness properties of the  functors on () and ().

<ref>. The theorem on level.
We introduce the notion of level, and prove a number of simple results concerning how our various functors interact with it. We then study a notion of saturation, and the effect of concatenating with the simple object ^1. After this, we prove the extremely important theorem on level, which states that any proper quotient of a rank r standard object has level <r. We close the section by giving some consequences of the theorem, e.g., that the standard modules generate the derived category (this is the first of the main results listed in <ref> to be proved).

<ref>. Grothendieck groups, Hilbert series, and Krull dimension.
We prove our main theorem on Grothendieck groups, namely, that the classes of standard modules form a basis. The theorem on level implies that they span; to prove linear independence, we use the plethora of functors available to us at this point to construct sufficiently many linear functionals to distinguish the classes in question. Using this description of the Grothendieck group, we give simple proofs of some facts about Hilbert series. We end with a brief discussion of Krull dimension.

<ref>. Induction and coinduction.
We define the induction and coinduction functors, which are the left and right adjoints of the shift functor. The induction functor is not so important, and we give it little attention. The coinduction functor, on the other hand, figures prominently in our study of injective objects, so we analyze it in some detail.

<ref>. Injective modules.
We prove our main results on injectives: namely, we classify the indecomposable injectives, and prove that finitely generated -modules have nice injective resolutions. To do this, we first construct the finite length injective objects, which is easy to do using coinduction. Then, using a combination of coinduction and some of our other functors, we show how to explicitly produce injectives that do not have finite length, and we produce enough of them to complete the classification. From the explicit form of these injectives, it is easy to write down an injective resolution of standard modules, from which it follows that they have finite injective dimension. Since they generate the derived category, it follows that all finitely generated modules have finite injective dimension.

<ref>. Local cohomology and saturation.
We review the general theory of saturation and local cohomology introduced in <cit.>, and then specialize it to -modules. Using the injective resolutions of standard modules found in the previous section, we compute the derived satuation and local cohomology of standard modules; it turns out that all the higher derived functors vanish in this case. We then introduce a new concept: we say that a module is r-semistandard if it admits a filtration where the graded pieces are rank r standard objects. Using our computations of local cohomology, we show that the r-semistandard objects form an abelian subcategory, which is rather surprising. Finally, we show that these abelian subcategories yield a semi-orthogonal decomposition of the derived category.

<ref>. Structure of level categories.
We explicitly identify the structure of the level categories (i.e., the graded pieces of the level filtration). The key input here is the classification of injective modules.

<ref>. Koszul duality.
In this section, we prove our main results about projective resolutions. We begin by defining the Tor functor on graded -modules and introducing the notion of a minimal projective resolution. We then construct the Koszul complex, which is a very natural complex that computes Tor. We then show that the definition of the Koszul complex can be modified to produce a derived auto-equivalence  of (); this is rather formal. We then prove the much more significant theorem that  preserves the bounded and finitely generated derived category. The proof of this theorem is quite short given the tools available to us at this point: since the standard modules generate the derived category, it suffices to show that (^λ) belongs to the bounded and finitely generated derived category; since  is compatible with concatenation produces, the task is further reduced to the case where λ is a word of length 1, where it can be handled directly. We then introduce the Betti table of a graded -module and deduce finiteness results about it from those of .

<ref>. Grothendieck groups revisited.
In this final section, we prove some deeper results about the Grothendieck group. We begin by defining the -pairings on () and (), which exist thanks to our results on injective resolutions. Next, we define the invariant μ_λ, which is, in a sense, the multiplicity of ^λ in an -module. We use these invariants to define the non-commutative Hilbert series _M, which we prove to be rational. We also introduce some variants of _M, which can be proved to be rational by the same methods, and yield a finite procedure for computing the pairings on the Grothendieck group. We close by determining the effective cone in the Grothendieck group.

Appendix <ref>. Categorical background.
Here we collect a variety of abstract results on abelian categories.



 §.§ Notation


We collect here some of the most important notation introduced in the body of the paper. This list is not meant to be exhaustive.


  * ^: the category of finite length objects in the abelian category .

  * ^: the category of locally finite objects in the abelian category .

  * ^: the category of finitely generated objects in the abelian category .

  * (): the derived category of the abelian category .

  * ^b_(): the bounded and finitely generated derived category of .

  * : the set of non-negative integers.

  * _+: the set of positive integers.

  * _n: the symmetric group on n letters.

  * : the increasing monoid (<ref>).

  * α_i, for i ≥ 1: one of the basic generators of  (<ref>).

  * _≥ n: the submonoid of  generated by the α_i with i ≥ n (<ref>).

  * : the coefficient field (fixed throughout).

  * (): the category of all -modules (<ref>).

  * (): the category of smooth -modules (<ref>).

  * (): the category of graded -modules (<ref>).

  * _n: the ideal of [] generated by the α_i-1 with i ≥ n (<ref>).

  * Φ: the forgetful functor () →() (<ref>).

  * ^r: the rth principal module (<ref>).

  * ^n: the nth simple module (<ref>).

  * ^λ: the standard module associated to λ (<ref>).

  * (-): initial term or module, in the sense of Gröbner theory (<ref>).

  * ⊙: the concatenation product (<ref>).

  * Σ: the shift functor (<ref>).

  * M^†: the transpose of the graded -module M (<ref>).

  * μ_n(M): the multiplicity of ^n in M (<ref>).

  * τ^<n, τ_≥ n: the truncation functors (<ref>).

  * Ψ: the direct limit functor () →() (<ref>).

  * Γ: the right adjoint of Ψ, given by M ↦⊕_n ≥ 0 M^_≥ n (<ref>).

  * Ξ: the completion functor, left adjoint of Φ (<ref>).

  * ()_≤ r: the level ≤ r category (<ref>).

  * (): the Grothendieck group of ()^ (<ref>).

  * _M: the Hilbert series of M (<ref>).

  * : the co-induction functor (<ref>).

  * : the induction functor (<ref>).

  * ^n: the injective envelope of the simple ^n (<ref>).

  * ^λ: the injective module associated to λ (<ref>).

  * _≤ r: the local cohomology functor with respect to ()_≤ r (<ref>).

  * _>r: the saturation functor with respect to ()_≤ r (<ref>).

  * _i: the ith Tor functor (<ref>).

  * (M): the Koszul complex on M (<ref>).

  * : the duality functor (<ref>).

  * ⟨,⟩: the pairing on () (<ref>).

  * μ_λ: the higher multiplicity invariant (<ref>).

  * _M: the non-commutative Hilbert seires of M (<ref>).

We typically use an underline to denote something in the graded setting, and no underline to denote something in the smooth setting. For example, ^λ is the standard module in the graded category and ^λ is the standard module in the smooth category; similarly, Σ is the shift functor on graded modules and Σ is the shift functor on smooth modules.



 §.§ Acknowledgements


We thank Steven Sam for helpful discussions.



§ THE INCREASING MONOID
 



 §.§ The increasing monoid
 

The big increasing monoid, denoted ^ big, is the monoid of all order-preserving injective maps _+ →_+, where _+ denotes the set of positive integers. We say that σ∈^ big is tame if there exists an integer ℓ≥ 0, called the length of σ, such that σ(n)=n+ℓ for all n ≫ 0. The small increasing monoid, denoted , is the submonoid of ^ big consisting of all tame elements. One easily verifies that it is indeed a submonoid, and that the function ℓ→ assigning to each element its length is a monoid homomorphism. For the questions studied in this paper, it turns out not to matter which version of the increasing monoid one uses, as we explain in <ref>. We will therefore always work with the small version .


The two versions of the increasing monoid are analogous to the two versions of the infinite symmetric group: the big version _∞^ big is the group of all bijections of _+, while the small version _∞ is the union of the _n. The length homomorphism on  is analogous to the sign homomorphism on _∞.



Given σ∈^ big, we extend it to a function → by declaring σ(0)=0.




 §.§ The generators
 

For k ≥ 1, let α_k denote the element of  defined by

α_k(n) = 
n+1     if n ≥ k 

n     if n<k. 

Clearly, ℓ(α_k)=1. One can show that the α_k are the only elements of length 1; in fact, this follows from Proposition <ref> below.

The following proposition gives the basic relations satisfied by the α's. We will use it constantly throughout the paper:

 
For n>m≥ 1 we have α_n α_m=α_m α_n-1.



This is a simple verification that is left to the reader.


 
Let σ∈.


  * There exists a unique strictly increasing sequence of integers 1 ≤ n_1<n_2<⋯<n_r such that σ=α_n_r⋯α_n_1.

  * There exists a unique weakly increasing sequence of integers 1 ≤ m_1 ≤ m_2 ≤⋯≤ m_s such that σ=α_m_1⋯α_m_s.

Moreover, r=s is the length of σ.



(a) Suppose that we have a decomposition σ=α_n_r⋯α_n_1 with n_r>n_r-1>⋯>n_1 ≥ 1. We then have:

σ(k) = 
k     if 1 ≤ k < n_1 

k+1     if n_1 ≤ k < n_2-1 

k+2     if n_2-1 ≤ k < n_3-2 

⋮    ⋮

k+r     if n_r-r+1 ≤ k


We can thus recover the n's from σ, and so we have uniqueness.

Conversely, we can start with σ and define the n's so that the above holds: for example, n_1 is the minimal value of k for which σ(k)>k. With these n's, we then have σ=α_n_r⋯α_n_1, and so we have existence.

(b) The proof is similar to (a), and the details are left to the reader.


 
The α's generate , and the relations in Proposition <ref> generate all relations among the α's.



The proposition obviously implies that the α's generate . Since the fundamental relations are sufficient to transform any word in the α's into a word of the type appearing in Proposition <ref>(a) (or Proposition <ref>(b)), it follows that they generate all relations.




 §.§ Smooth actions
 

Let ^ big_>n be the submonoid of ^ big consisting of those elements σ that fix each of the numbers 1, …, n. Let _>n=∩^ big_>n. Clearly, _>n is the submonoid of  generated by the α_k with k>n.


Let X be an -set. We say that an element x ∈ X is smooth if x is fixed by _>r for some r. We say that X is smooth if every element is.


We make a similar definition for ^ big. We almost exclusively study smooth representations in this paper. The following proposition thus explains why we could work with either version of the increasing monoid:


The restriction functor

{ smooth ^ big-sets } →{ smooth -sets }

is an isomorphism of categories.



Let X be a smooth -set. We claim that there is a unique smooth action of ^ big on X extending that of . To see this, suppose that σ∈^ big and x ∈ X. Let r be such that x is fixed by _>r. We can then factor σ as σ_1 σ_2 where σ_1 ∈ and σ_2 ∈^ big_>r: indeed, we take σ_1(n)=σ(n) for 1 ≤ n ≤ r, and then put σ_1(n)=n+ℓ for n ≥ r, where ℓ=σ(r)-r; we then take σ_2(n)=1 for 1 ≤ n ≤ r and σ_2(n)=σ(n)-ℓ for n ≥ r+1. We define σ x to be σ_1 x. We leave to the reader the routine verification that this does indeed define a smooth action of ^ big on X. To verify uniqueness, suppose that ∙ is a second smooth action of ^ big on X extending the action of . Let σ and x be as above, let r be such that x is fixed by ^ big_>r under the action ∙ (and thus by _>r), and factor σ as σ_1 σ_2 as above. Then σ∙ x = σ_1 ∙ x = σ_1 x = σ x, which shows that ∙ agrees with the previously defined action.

The above construction defines a functor

{ smooth -sets } →{ smooth ^ big-sets }.

It is clear that ∘ and ∘ are both equal to the identity functor, and so  and  are mutually inverse isomorphisms of categories.


The following proposition and corollary give a convenient criterion for an element of an -set to be smooth.

 
Let X be an -set and let x ∈ X. Suppose α_n x = x. Then x is fixed by all of _≥ n.



Suppose α_i x = x. We have the identity α_i+1α_i = α_i^2 in  by Proposition <ref>. Applying both sides to x, we find α_i+1 x = x. We thus see that x is fixed by α_i for all i ≥ n, and these generate _≥ n.



An element of an -set is smooth if and only if it is fixed by some α_n.




§ REPRESENTATION CATEGORIES
 



 §.§ The category of all -modules
 

Fix, for the remainder of the paper, a field . We let () be the category of all representations of  over . Thus () is simply the category of left modules over the monoid algebra []. As such, it is a Grothendieck abelian category. This category will only play a tangential role in this paper.



 §.§ The category of smooth -modules
 

Let () be the full subcategory of () spanned by smooth modules. As any subquotient of a smooth module is again smooth, we see that () is an abelian subcategory of ().

Let {M_i}_i ∈ I be a family of smooth -modules. It is clear that the coproduct ⊕_i ∈ I M_i (taken in the category of all -modules) is again smooth, and is thus the coproduct in (). It follows that arbitrary colimits in () can be computed in (). We thus see that () is cocomplete and that filtered colimits in () are exact. It is clear that () has a generator: simply take the direct sum of all smooth cyclic -modules. (Or, more accurately, one from each isomorphism class. It is clear that the collection of all isomorphism classes forms a set.) We thus see that () is a Grothendieck abelian category.

For an -module M, we let M^ be the set of all smooth elements in M; it is easily verified to be an -submodule of M, and is clearly the maximal smooth submodule. Explicitly,

M^ = ⋃_r ≥1 M^_≥r = _r →∞ M^_≥r.

Here M^_≥ r denotes the subspace of M consisting of elements invariant under _≥ r. One easily sees that M ↦ M^ is right adjoint to the inclusion functor () →().

Let {M_i}_i ∈ I be a diagram of smooth -modules. The limit M_i, as computed in (), need not be smooth. We define the smooth limit, denoted ' M_i, to be the maximal smooth submodule of M_i. Explicitly,

_i ∈I' M_i
= _r →∞ [_i ∈I M_i ]^_≥r
= _r →∞ [_i ∈I M_i^_≥r ]

In the second step, we have used the fact that formation of invariants commutes with limits. One readily verifies that ' M_i is the limit of the diagram {M_i} in the category (). When the category I is discrete, so that limits over I are just products, we write ∏'_i ∈ I M_i for the smooth limit, and refer to it as the smooth product. One has the explicit description

∏_i ∈I'  M_i = _r →∞ [ ∏_i ∈I M_i^_≥r ]

Of course, the smooth product is the product in the category ().


Products in () are not exact, that is, Grothendieck's (AB4*) axiom does not hold.



For an integer n ≥ 1, let M_n=⊕_k ≥ n e_k, with  acting by σ e_k=e_σ(k). This is a smooth -module. Let ϵ_n  M_n → be the augmentation map (i.e., the linear map sending each e_i to 1), which is a surjection of -modules (giving  the trivial action). We claim that ∏'_n ≥ 1ϵ_n is not surjective. Indeed, suppose x=(x_n)_n ≥ 1 is an element of ∏'_n ≥ 1 M_n. Then, by definition, x belongs to ∏_n ≥ 1 M_n^_≥ r for some r. Note that M_n^_≥ r=0 for n ≥ r. Thus x_n=0 for n ≥ r, and so ϵ_n(x_n)=0 for n ≥ r as well. We thus see that the image of ∏'_n ≥ 1ϵ_n is contained in ⊕_n ≥ 1, which is a proper submodule of ∏'_n ≥ 1=∏_n ≥ 1.


Due to the proposition, we must take care when dealing with the derived functors of lim': for example, we are not guaranteed that ^i lim' vanishes for i>1, or that Mittag–Leffler systems are lim'-acyclic.



 §.§ The category of graded -modules
 

A graded -module is an -module M equipped with a decomposition M=⊕_n ≥ 0 M_n such that two conditions hold:


  * Given x ∈ M_n and σ∈, the element σ x belongs to M_σ(n).

  * Given x ∈ M_n and σ∈ such that σ(n)=n, we have σ x = x.

By convention, every element of  fixes 0; thus every degree 0 element of a graded -module is -invariant. We let () denote the category of graded -modules. We use the notation _ for the  sets in (); that is, for M,N ∈(), we let _(M,N) denote the space of all homogeneous -linear maps M → N. If M is a graded -module then the action of  on M is smooth by condition (b). We thus have a forgetful functor

Φ() →().


 
The forgetful functor Φ is continuous and cocontinuous.



Since the forgetful functor is exact, it suffices to show that it commutes with products and co-products. Co-products is clear: for both graded and smooth modules, corproducts are computed using the coproduct of the underlying vector space. We now treat products. Let {M_i}_i ∈ I be a family of graded -modules. We have

( ∏_i ∈I [ ⊕_n ≥0 M_i,n ] )^_≥r
= ∏_i ∈I [ ⊕_0 ≤n < r M_i,n ]
= ⊕_0 ≤n < r [ ∏_i ∈I M_i,n ].

In the first equality, we used the fact that taking invariants commutes with products and coproducts, as well as the fact that M_i,n^_≥ r vanishes for n ≥ r, and equals M_i,n for n<r. In the second step, we used the fact that products commute with finite coproducts. Taking the union over r, we now find

∏_i ∈I' [ ⊕_n ≥0 M_i,n ] = ⊕_n ≥0 [ ∏_i ∈I M_i,n ].

The left side computes the result of first applying the forgetful functor and then taking products, while the right side computes the result of first taking products and then forgetting. The result follows.


Let M be a graded -module. We let M_≥ n be the subspace that is M_k in degree k ≥ n and 0 in degree k<n. This is clearly a homogeneous -submodule of M. We let M^<n=M/M_≥ n. We also put M_+=M_≥ 1. We say that M is pure if M_0=0, and denote the category of pure modules by ()_+.


Since elements of  fix 0, the degree 0 part of a graded -module cannot interact with the positive degree part. In other words, the category () decomposes as ⊕()_+. It may therefore seem somewhat odd to include the degree 0 piece in the definition. However, we will see several instances in which having the degree 0 part is more natural; see, for example, Remarks <ref> and <ref>.




 §.§ Coinvariants
 

Let M be a an -module. We write M__≥ n for the coinvariant space under _≥ n. This is the maximal quotient of M on which _≥ n acts trivially; explicitly, it is the quotient of M by the subspace spanned by elements of the form σ x -x with σ∈_≥ n and x ∈ M. Let _n be the right ideal of [] generated by the elements σ-1 with σ∈_≥ n. Thus M__≥ n=M/_n M.

 
The right ideal _n is a two-sided ideal of []. It is generated, as a left or right ideal, by the elements α_i-1 with i ≥ n. 



We first show that _n is generated as a right ideal by the α_i-1 with i ≥ n. Let '_n be the right ideal generated by the α_i-1 with i ≥ n. We obviously have '_n ⊂_n; we prove the reverse inclusion. Let σ∈_≥ n. It suffices to show that σ-1 ∈_n'. Since the α_i with i ≥ n generated _≥ n, we can write σ=τα_i where i ≥ n and τ∈_≥ n has shorter length than σ. We have the identity

σ-1=(τ-1) α_i+(α_i-1).

We can assume, by induction on length, that τ-1 belongs to '_n. Since α_i-1 belongs to '_n by definition, we thus find that σ-1 ∈'_n, as required.

We now show that _n is a two-sided ideal. It suffices to show that α_j (α_i-1) ∈_n for all i ≥ n and all j ≥ 1. First suppose that j ≤ i. Then

α_j (α_i-1) = (α_i+1-1) α_j,

and thus belongs to _n, as i+1 ≥ n. Now suppose j>i. Then

α_j (α_i-1) = α_i α_j-1 - α_j
=(α_i-1) α_j-1+(α_j-1-1)-(α_j-1),

and thus belongs to _n, as i, j, and j-1 are all ≥ n.

Finally, let _n” be the left ideal generated by the α_i-1 for i ≥ n. Since _n is a left ideal containing these elements, we have _n”⊂_n. An argument as in the previous paragraph shows that _n” is a two-sided ideal. Since _n is the smallest right ideal containing the elements α_i-1 for i ≥ n, it is also the smallest two-sided ideal containing these elements, and so _n”=_n.



Let M be an -module. Then _n M is an -submodule of M. Thus M__≥ n is canonically an -module and the canonical map M → M__≥ n is -linear.




 §.§ -modules
 

Recall that  is the category whose objects are finite totally ordered sets and whose morphisms are order-preserving injections. For n ∈, let [n] be the set {1,…,n} equipped with its usual total order. Every object of  is isomorphic to a unique [n]. For 1 ≤ i ≤ n+1, let β_n,i [n] → [n+1] be the map defined by

β_n,i(j) = 
j     if j<i 

j+1     if j ≥ i


These are all the morphisms from [n] to [n+1] in the category . Moreoever, every morphism in  can be realized as a composition of these morphisms. We have the relations

    β_n+1,i∘β_n,j = β_n+1,j∘β_n,i-1

for 1 ≤ j < i ≤ n+2, and these relations generate all relations between the β's. The above claims can be proved similarly to Corollary <ref>; they are also similar to standard facts about the simplex category, see <cit.>.

 
We have an equivalence of categories ()_+ ≅_.



Let M be an -module. We define a graded -module N as follows. As a graded vector space, N_0=0 and N_n=M([n-1]) for n ≥ 1. For 1 ≤ i ≤ n, we define α_i  N_n → N_n+1 to be the map β_n-1,i M([n-1]) → M([n]). The fundamental relations correspond exactly to the identities (<ref>). This construction is reversible: starting with N we can define M. It thus furnishes the claimed equivalence.




 §.§ Semi-simplicial vector spaces
 

Let Δ be the simplex category: its objects are finite non-empty totally ordered sets and its morphisms are order-preserving functions. Let Δ_ inj be the subcategory of Δ with the same objects but where the morphisms are injective. Recall that a simplicial object in a category  is a functor Δ^→, and that a semi-simplicial object in  is a functor Δ_ inj^→. One also defines co-simplicial and co-semi-simplicial objects in  as functors Δ→ and Δ_ inj→. For general background on simplicial objects, see <cit.>.

The Dold–Kan theorem <cit.> states that the category of simplicial objects in an abelian category  is equivalent to the category of chain complexes in  supported in non-negative homological degrees. Since chain complexes of vector spaces are the same as graded modules over the ring [ϵ]/(ϵ^2), one can give very precise results on their structure. Thus, by the Dold–Kan theorem, simplicial vector spaces have a fairly simple structure. Semi-simplicial vector spaces, however, are much more complicated. The following proposition shows that the results of this paper can be used to understand them:


The category of co-semi-simplicial vector spaces is equivalent to the full subcategory of () spanned by graded -modules M with M_0=M_1=0.



The category Δ_ inj is simply the full subcategory ' of  spanned by non-empty sets. An '-module M can be extended to an -module by simply putting M(∅)=0, and this construction is an equivalence between the category of '-modules and the category of -modules M with M(∅)=0. The result now follows from Proposition <ref>.



Semi-simplicial and co-semi-simplicial vector spaces are not so different: if M is a co-semi-simplicial vector space then S ↦ M(S)^* is a semi-simplicial vector space, and this construction is an equivalence on the categories of (co-)semi-simplicial objects taking values in finite dimensional vector spaces.




 §.§ Shuffle algebras
 

Given n,m ∈, an (n,m)-shuffle is a bijection [n] ⨿ [m] → [n+m] that is order-preserving on each summand. Here, [n] denotes the set {1,…,n}. Let (n,m) denote the set of (n,m) shuffles. Given -graded vector spaces V and W, define a new graded vector space V  W by

(V W)_n = ⊕_i+j=n [(i,j)] ⊗V_i ⊗W_j.

The construction  endows the category  of graded vector spaces with a symmetric monoidal structure. The associativity constraint comes from the natural bijection

(n,m) ×(n+m, ℓ) ≅(n,m+ℓ) ×(m,ℓ),

while the symmetry comes from the natural bijection

(n,m) ≅(m,n).

A (commutative) shuffle algebra is a (commutative) associative unital algbera in the symmetric monoidal category (, ). More explicitly, a shuffle algebra is a graded vector space A=⊕_n ≥ 0 A_n equipped with, for each shuffle σ∈(n,m), a multiplication ∗_σ A_n ⊗ A_m → A_n+m, satisfying a number of axioms.

The monoidal operation  on  and shuffle algebras are discussed in <cit.>, and also <cit.>, to which we refer for further details. Shuffle algebras also appear in the works of Sam <cit.> and Laudone <cit.> that study syzygies of certain families of varieties. Our interest in shuffle algberas comes from the following observation:

 
Let ⟨ 1 ⟩ denote the graded vector space that is  in degree 1, and 0 in other degrees. Let A be the shuffle algebra _(⟨ 1 ⟩). Then the category _A of left A-modules is equivalent to the category _.



Since this result is only used as motivation, we simply sketch the proof. Given a shuffle σ∈(n,m), let f(σ)  [m] → [n+m] be the restriction of σ to [m], regarded as a morphism in . Observe that

f (n,m) →_([m], [n+m])

is a bijeciton.

Now, let M be a graded vector space. To give an A-module structure on M requires giving a multiplication map ∗_σ A_n ⊗ M_m → M_n+m for each σ∈(n,m). However, A_n is one-dimensional, so this is the same as giving a map M_m → M_n+m for each element of (n,m). By the above bijection, this is the same as giving a map M_m → M_n+m for each element of _([m], [n+m]), which is exactly the data required to give M the structure of an -module. One must now verify that the conditions on the data on either side match, which we leave to the reader.


 
We have an equivalence of categories _A ≅()_+.



Simply combine the proposition with the equivalence ()_+ ≅_ (Proposition <ref>).


 
One can show that the category of left modules over the shuffle algebra _(⟨ 1 ⟩^⊕ d) is equivalent to the category __d of _d-modules, where _d is the generalization of  defined in <cit.>.



The shuffle algebra perspective on -modules brings great clarity to the work in <ref> on Koszul duality. See <ref> for details.




§ MONOMIAL MODULES
 



 §.§ Monomial modules


A pointed -set is a set  equipped with an action of  and a distinguished point ∗ (called zero) that is fixed by . Let  be a pointed set. We let _+=∖{∗}. Wet [] be the vector space with basis _+, and for i ∈_+, we let e_i ∈[] be the corresponding basis vector. We also put e_∗=0. The space [] carries an action of  via σ e_i=e_σ(i). If the action of  on  is smooth then the action of  on [] is also smooth. We call modules of this form monomial modules, and we refer to the elements e_i as monomials.

Let  be a pointed -set. A grading on  is a function δ→ that is -equivariant and satisfies the following condition: if i ∈ and σ∈ is such that σ stabilizes δ(i) then σ stabilizes i. Given a grading on , we give [] the structure of a graded -module by declaring e_i to have degree δ(i).



 §.§ Principal modules
 

Let ^r_+ be the set consisting all strictly increasing tuples in _+^r, and let ^r=^r_+ ∪{∗}. This set is naturally a smooth pointed -set. We let ^r be the associated monomial module. We call these principal modules. For 1 ≤ i_1<⋯<i_r, we let e_i_1,…,i_r denote the corresponding basis vector of ^r. The set ^r admits a grading, defined by δ(i_1,…,i_r)=i_r. We let ^r denote the corresponding graded -module. The basis vector e_i_1,…,i_r of ^r has degree i_r.


Let M be an arbitrary -module. Then _(^r, M) is canonically isomorphic to M^_>r. Precisely, if x ∈ M^_>r then there is a unique -linear map f ^r → M such that f(e_1,…,r)=x.



Let x ∈ M^_>r be given. Suppose that σ,τ∈ satisfy σ(e_1,…,r) =τ(e_1,…,r). Thus σ(i)=τ(i) for 1 ≤ i ≤ r. Thus if σ=α_1^n_1α_2^n_2⋯ and τ=α_1^m_1α_2^m_2⋯ then n_i=m_i for 1 ≤ i ≤ r; indeed, σ(1)=n_1, σ(2)=n_1+n_2, and so on. It follows that σ x = τ x. Indeed, since x is fixed by the α_k with k>r, we have σ x = α_1^n_1⋯α_r^n_r x = τ x. It follows that we have a well-defined -equivariant map ^r ∖{∗}→ M by σ (1,…,r) ↦σ x for σ∈. This extends linearly to an -linear map f ^r → M satisfying f(e_1,…,r)=x. Since e_1,…,r generates ^r, it follows that f is the unique map taking e_1,…,r to x.


 
Let M be a smooth -module. Then M is a quotient of a direct sum of principal modules. If M is finitely generated, then it is a quotient of a finite direct sum of principle modules.



Let M be a smooth -module and let {x_i}_i ∈ I be a set of generators for M. Since M is smooth, each x_i is fixed by some _>r_i. We thus have a unique map f_i ^r_i→ M that sends e_1,…,r_i to x_i. Since the x_i generate M, the map ⊕_i ∈ I f_i ⊕_i ∈ I^r_i→ M is surjective.



Let M be a graded -module. Then _(^r, M) is canonically isomorphic to M_r. Precisely, if x ∈ M_r then there is a unique homogeneous -linear map f ^r → M such that f(e_1,…,r)=x.



Let x ∈ M_r be given. Since x is _>r invariant, the previous proposition yields a unique -linear map f ^r → M satisfying f(e_1,…,r)=x. Thus f(σ e_1,…,r)=σ x. Since both σ e_1,…,r and σ x have degree σ(r), and the σ e_1,…,r span ^r, it follows that f is homogeneous. This proves the proposition.



The graded principal modules are projective. Furthermore, every (finitely generated) graded -module is a quotient of a (finite) direct sum of graded principal modules.



The proposition shows that the functor _(^r, -) is isomorphic to the functor M ↦ M_r, and is thus exact; therefore ^r is projective. The second statement is clear.


A monomial submodule of a monomial -module is one that is spanned (over ) by the monomials it contains. The following proposition classifies the monomial submodules of ^r, and establishes an important link to commutative algebra.

 
Let r ∈. Consider the linear map f ^r →[x_1, …, x_r] defined by

f(e_i_1,…,i_r) = x_1^i_1-1 x_2^i_2-i_1-1 ⋯x_r^i_r-i_r-1-1.

Then f is an isomorphism of vector spaces. Moreover, f induces a bijection

{ monomial submodules of ^r } →{ monomial ideals of [x_1,…,x_r] }.




It is clear that f induces a bijection between the set of basis vectors in ^r and the set of monomials in [x_1,…,x_r], and is therefore an isomorphism of vector spaces. To prove the second statement, it suffices to show that if V ⊂^r is a subspace spanned by some collection of basis vectors then V is an -submodule of ^r if and only if f(V) is an ideal of [x_1, …, x_r]. Observe that we have the identity:

x_k f(e_i_1,…,i_r) = f(e_i_1,…,i_k-1,i_k+1,…,i_r+1).

From this, we see that the following two conditions are equivalent:


  * f(V) is an ideal in [x_1,…,x_r].

  * If e_i_1,…,i_r belongs to V then so does e_i_1,…,i_k-1,i_k+1,…,i_r+1, for any 1 ≤ k ≤ r.

It is clear that (b) is equivalent to α_j V ⊂ V for all j, and thus to V being an -submodule. This completes the proof.




 §.§ Simple modules
 

For n ∈, let ^n be the set {∗, n}. We give ^n the structure of a pointed -set by:

α_i ·n = 
∗    if i ≤ n 

n     if i>n


As such, it is smooth. The set ^n admits a grading, in which n is given degree n.

We let ^n be the monomial module associated to ^n, and ^n the graded version. These are one-dimensional, and thus simple. We note that ^0 is the 1-dimensional trivial representation, and thus isomorphic to ^0. The module ^n is pure for n>0.



 §.§ Standard modules
 

A constraint word a finite word λ=λ_1 ⋯λ_k in the 2-letter alphabet {, }. Fix such λ. Let ^λ_+ be the set of strictly-increasing tuples (n_1, …, n_k) with n_i ∈_+ such that n_i-n_i-1=1 if λ_i=; here we use the convention n_0=0. If λ is the empty word (i.e., k=0) then ^λ_+ is the singleton set consisting of the empty tuple. We let ^λ=^λ_+ ∪{∗}. We let -act on this set by:

σ(n_1, …, n_k) = 
(σn_1, …, σn_k)     if this belongs to ^λ_+ 

∗    otherwise.


Of course, we also put σ∗=∗ for all σ. In this way, ^λ has the structure of a smooth pointed -set. It is also admits a natural grading, by declaring (n_1, …, n_k) to have degree n_k. (For k=0, the empty singleton is given degree 0.)

We let ^λ be the monomial module associated to ^λ, and let ^λ be the graded version. We call these standard modules. A few remarks:


  * If λ=^k is the length k word with λ_i= for all i then ^λ≅^k and ^λ≅^k.

  * If λ=^k is the length k word with λ_i= for all i then ^λ≅^k and ^λ≅^k.

  * If λ is the empty word then ^λ≅^0 ≅^0 is the trivial representation. Otherwise, ^λ is pure.

We define the rank of λ to be the number of times  occurs in it. We define the rank of ^λ and ^λ to be that of λ. Thus, for example, ^r has rank r, while ^n has rank 0.



§ GRÖBNER THEORY
 



 §.§ Principal modules


Recall that the principal module ^r has a basis indexed by the set ^r_+ of increasing sequences (i_1,⋯,i_r) of positive integers. Let < be the lexicographic order on ^r_+, where the larger numbers are compared first; to be precise, (i_1,…,i_r)<(j_1,…,j_r) if i_k<j_k, where k is the maximal index such that i_k  j_k. The order < is a well-order on ^r_+, and is compatible with the -action in the sense that (i_1,…,i_r)<(j_1,…,j_r) implies σ (i_1,…,i_r) < σ (j_1, …, j_r) for any σ∈.

Given a non-zero element x=∑_(i_1,…,i_r) ∈^r_+ c_i_1,…,i_r e_i_1,…,i_r of ^r, we define the initial term of x, denoted (x), to be e_i_1,…,i_r, where (i_1,…,i_r) is the maximal element of ^r_+ with c_i_1,…,i_r non-zero. Since  is compatible with the order, we see that (σ x)=σ(x) for any σ∈.

Now let M be an -submodule of ^r. We define the initial submodule, denoted (M), to be the the subspace of ^r spanned by the elements (x) for x ∈ M. This is a monomial submodule of ^r: indeed, it is a submodule since σ(x)=(σ x), and it is monomial since it is spanned by a collection of basis vectors. We have the usual Gröbner lemma:

 
Suppose that M ⊂ N are -submodules of ^r such that (M)=(N). Then M=N.



Suppose that M  N, and choose x ∈ N ∖ M with (x) minimal; this is possible since < is a well-order on ^r_+. Since (N)=(M), we have (x)=(y) for some y ∈ M. There is thus a non-zero scalar c such that (x-cy)<(x). By the minimality of x, we see that x-cy ∈ M. Since cy ∈ M, this implies x ∈ M, a contradiction. Thus M=N.


The main application of the above material is the following theorem:

 
The categories () and () are locally noetherian.



We first claim that ^r is a noetherian object of (), for any r. Indeed, suppose that M_1 ⊂ M_2 ⊂⋯⊂^r is an ascending chain of -submodules. We then have an ascending chain (M_1) ⊂(M_2) ⊂⋯ of monomial submodules of ^r. By Proposition <ref>, this corresponds to an ascending chain of monomial ideals in the polynomial ring [x_1,…,x_r], and thus stabilizes since this ring is noetherian. (It is also easy to directly prove that this chain stabilizes.) By Proposition <ref>, it follows that the original chain M_∙ stabilizes, and so ^r is noetherian.

It now follows that ^r is a noetherian object of (): indeed, an ascending chain of homogeneous submodules of ^r is an ascending chain of submodules of ^r that happens to also be homogeneous, and so stabilizes.

Any subquotient of a noetherian object is noetherian; also, any finite direct sum of noetherian objects is noetherian. Since every finitely generated object in () is a quotient of a finite sum of principal modules (Corollary <ref>), it follows that all finitely generated objects of () are noetherian. Thus () is locally noetherian. The same reasoning applies to ().



The local noetherianity of () was previously known: in <cit.>, it is shown that the category _ of -modules is locally noetherian (also using a Gröbner-theoretic argument), which implies local noetherianity of () via the equivalence in Proposition <ref>. The noetherian result for () has certainly been known to experts in the field for some time, but we do not know if it previously appeared in the literature.




 §.§ Graded modules
 

We now develop a version of Gröbner theory that applies to inhomogeneous submodules of graded modules. We use this theory here to deduce a few properties of the forgetful functor Φ. We will also use it later in our study of saturation (see Proposition <ref>).

Let M be a graded -module. We will regard M as an object of () in what follows. Given a non-zero element x ∈ M, let x=∑_i ∈ x_i be its decomposition into homogeneous pieces. We define the initial term of x, denoted (x), to be x_i where i is maximal so that x_i is non-zero; we also define the degree of x, denoted (x), to be this value of i. We define the initial submodule of a (possibly inhomogeneous) -submodule N of M, denoted (N), to be the subspace of M spanned by the elements (x) as x varies over N; it is a homogeneous -submodule of M. We say that x_1, …, x_n ∈ N are a Gröbner basis if (x_1), …, (x_n) generate (N).

We shall require the following results on Gröbner bases. They are entirely standard, at least in the usual context of polynomial rings, but we include proofs to be complete.


Let M be a finitely generated graded -module, and let N be a (possibly inhomogeneous) -submodule. Then N admits a Gröbner basis.



Suppose not. Inductively define a sequence x_1, x_2, … in N by taking x_n+1 to be any element of N such that (x_n+1) does not belong to the submodule of M generated by (x_1), …, (x_n); note that such x_n+1 exists since x_1, …, x_n are not a Gröbner basis. We thus see that the submodule of M generated by {(x_n)}_n ≥ 1 is not finitely generated, contradicting the noetherianity of M (Theorem <ref>).


 
Let M be a graded -module, let N be a (possibly inhomogeneous) -submodule, and let x_1, …, x_n be a Gröbner basis of N. Let y ∈ N. Then we have an expression

y = ∑_i=1^m c_i σ_i x_a_i

where c_i ∈, σ_i ∈, and (σ_i x_a_i) ≤(y) for all j.



If y=0 then the statement is obvious. Thus suppose y  0, and let us proceed by induction on d=(y). Since (y) belongs to (N), it can be generated by the homogeneous elements (x_1), …, (x_n). We therefore have an expression (y) = ∑_i=1^m c_i σ_i (x_a_i) for some scalars c_i and indices a_i. As the terms in this sum are homogeneous and the result has degree d, all terms not of degree d cancel, and we can therefore exclude them. We may thus assume (σ_i (x_a_i))=d for all i. Note that (σ_i x_a_i)=d as well. Let y'=y-∑_i=1^m c_i σ_i x_a_i. Then y' has degree <d. By the inductive hypothesis, we have an expression y'=∑_j=1^ℓ c'_j σ'_j x_a'_j where (σ'_j x_a'_j)<d for all j. We thus find y=∑_j=1^ℓ c'_j σ'_j x_a'_j-∑_i=1^m c_i σ_j x_a_i, as required.


 
Let M be a graded -module. Suppose that A and B are possibly inhomogeneous submodules of M such that M=A ⊕ B. Then M=(A) ⊕(B).



We first claim that M=(A)+(B). Thus let x ∈ M_k be given. Write x=a+b with a ∈ A and b ∈ B. Since x belongs to M_k, it is invariant under _>k, and so a and b are each invariant under _>k. Thus a,b ∈ M^_>k=∑_i=0^k M_i. Let a=∑_i=0^k a_i and b=∑_i=0^k b_i be the decompositions of a and b into homogeneous pieces. Thus x=a_k+b_k. As a_k is either 0 or (a), and similarly for b_k, the claim is established.

We now show that (A) ∩(B)=0. Suppose not, and say (a)=(b) is non-zero of degree k, where a∈ A and b ∈ B have degree k. Then a-b has degree <k, and so, by the previous paragraph, can be written in the form ∑_i=0^k-1[ (a_i) - (b_i) ], where the a_i belong to A and the b_i belong to B, and a_i and b_i have degree ≤ i. Let a'=a-∑_i=0^k-1 a_i and b'=b-∑_i=0^k-1 b_i. Then a'=b'. Since this belongs to A ∩ B=0, we find a'=0 and b'=0. But this is a contradiction, as (a')=(a) is non-zero. We conclude that (A) ∩(B)=0.


 
Let M be a graded -module. Then M is indecomposable if and only if Φ(M) is indecomposable.



It is clear that if Φ(M) is indecomposable then M is indecomposable. Suppose now that Φ(M) is decomposable, so that M=A ⊕ B for (possibly inhomogeneous) non-zero submodules A and B of M. By the proposition, we have M=(A) ⊕(B), and so M is decomposable.


 
An exact sequence of graded -modules is split if and only if it is split after applying Φ.



Consider an exact sequence of graded -modules:

0 →M_1 →M_2 →M_3 →0.

Obviously, if it is split then it remains split after applying Φ. Suppose that it splits after applying Φ. We can thus find a decomposition M_2=M_1 ⊕ K where K is a possibly inhomogeneous submodule of M_2. By Proposition <ref>, we have M_2=M_1 ⊕(K); note that since M_1 is homogeneous, we have (M_1)=M_1. Since (K) is a homogenous submodule of M_2, we see that the sequence splits in the graded category.




§ CONCATENATION, SHIFT, AND TRANSPOSE
 



 §.§ The concatenation product
 

Let M be a graded -module and let N be an arbitrary -module. We define a new -module, called the concatenation of M and N, and denoted M ⊙ N, as follows. As a vector space, M ⊙ N is just the usual tensor product M ⊗ N, where here M denotes the underlying ungraded vector space of M. For x ∈ M and y ∈ N, we write x ⊙ y in place of x ⊗ y when we regard it as an element of M ⊙ N; this will help us keep track of the -actions better. The action of  on M ⊙ N is defined as follows: for x ∈ M_n and y ∈ N, we put

α_k ·(x ⊙y) = 
(α_k x) ⊙y      if 1 ≤ k ≤ n   

x ⊙(α_k-n y)     if n<k


We leave to the reader the simple verification that this action is well-defined (i.e., the fundamental relations are respected). It is clear that M ⊙ N is smooth if N is. If N is graded then M ⊙ N is in fact a graded -module, under the usual grading (i.e., if x ∈ M_i and y ∈ N_j then x ⊙ y ∈ (M ⊙ N)_i+j). We note that, essentially by definition, we have Φ(M ⊙ N) = M ⊙Φ(N). One readily verifies that ⊙ defines a (non-symmetric) monoidal structure on (), with unit object ^0=^0, and also defines on () the structure of a module category over the monoidal category ().

For constraint words λ=λ_1 ⋯λ_r and and μ=μ_1 ⋯μ_s, let λ⊙μ be the concatenated word λ_1 ⋯λ_r ·μ_1 ⋯μ_s.

 
We have a natural isomorphism ^λ⊙^μ≅^λ⊙μ of graded -modules. Similarly, we have an isomorphism ^λ⊙^μ≅^λ⊙μ.



Let λ=λ_1 ⋯λ_r and μ=μ_1 ⋯μ_s. Define a map ^λ⊙^μ→^λ⊙μ on basis vectors by

e_i_1,…,i_r ⊙e_j_1,…,j_s ↦e_i_1,…,i_r,i_r+j_1,…,i_r+j_s.

It is clear that this is a bijection on basis vectors and compatible with the action of . It thus defines the requisite isomorphism.


 
Let λ=λ_1 ⋯λ_r be a constraint word. Then we have

^λ ≅^λ_1 ⊙⋯⊙^λ_r.

The ith factor on the right is ^1 if λ_i=, or ^1 if λ_i=.


 
The nature of the concatenation product is one piece of evidence that having graded -modules start in degree 0 is the best choice: the unit object for ⊙ is concentrated in degree 0. Thus, for example, ()_+ is not a monoidal category under ⊙.




 §.§ The shift functor
 

Let M be an -module. We define Σ(M) to be the pullback of M under the homomorphism → defined by α_i ↦α_i+1. The vector space underlying Σ(M) is just M; however, for clarity, for x ∈ M we denote the corresponding element of Σ(M) by x^♭. The action on Σ(M) can thus be described by the formula

α_i ·x^♭ = (α_i+1 ·x)^♭.

It is clear that if M is smooth then so is Σ(M), and so Σ defines an endofunctor of (). It is obvious that Σ is exact and cocontinuous. It is also continuous: indeed, an element x of an -module M is smooth if and only if x^♭∈Σ(M) is smooth, and so Σ commutes with the operation (-)^, and thus with smooth products since it obviously commutes with products.

We also define a version of the shift functor in the graded case. Let M be a graded -module. We define Σ(M) to be the following graded -module. As a graded vector space, Σ(M)_n=M_n+1, and as before, for x ∈ M_n+1 we denote by x^♭ the corresponding element of Σ(M)_n. The -module structure is defined just as before, i.e., α_i · x^♭=(α_i+1· x)^♭. The one thing to verify is that _>n acts trivially on Σ(M)_n. Thus suppose i>n, and let x^♭ be an element of Σ(M)_n, with x ∈ M_n+1. Then α_i · x^♭=(α_i+1· x)^♭=x^♭, since α_i+1 acts trivially on M_n+1. We thus see that Σ is a well-defined endofunctor of (). It is clearly exact, continuous, and cocontinuous. The graded and ungraded shifts are compatible on pure modules, that is, if M is a pure graded module then Φ(Σ(M))=Σ(Φ(M)). If M is a graded module concentrated in degree 0 then Σ(M)=0, while Σ(Φ(M))=Φ(M).

 
Suppose M is an -module, and let n ≥ 1. Then M^_≥ n=(Σ^n-1 M)^ and M__≥ n=(Σ^n-1 M)_.



This follows immediately from the definitions.


 
Let M be an -module. Then the function f  M →Σ(M) defined by f(x)=(α_1 · x)^♭ is -linear. Moreover, if M is graded then f is homogeneous.



We have

f(α_i x)=(α_1α_ix)^♭=(α_i+1 α_1 x)^♭=α_i(α_1 x)^♭=α_i f(x),

which proves the result.



 
Let M be a graded -module and let N be an arbitrary -module. Then we have a natural isomorphism

Σ(M ⊙N) ≅[ Σ(M_+) ⊙N ] ⊕[ M_0 ⊙Σ(N) ].

If N is graded, then this isomorphism is homogeneous.



Define

f Σ(M_+) ⊙N →Σ(M ⊙N),     f(x^♭ ⊙y)=(x ⊙y)^♭

and

g M_0 ⊙Σ(N) →Σ(M ⊙N),     g(x ⊙y^♭)=(x ⊙y)^♭.

Then f ⊕ g is an isomorphism of vector spaces: the source are target are both identified with M ⊗ N, and under this identification f ⊕ g is simply the identity map. Moreover, if N is graded then f and g are homogeneous. We verify that f and g are -linear.

Suppose x ∈ M_i+1 and y ∈ N. Then

α_k ·(x^♭ ⊙y) = 
(α_k+1 x)^♭ ⊙y     if 1 ≤ k ≤ i 

x^♭ ⊙(α_k-i y)     if i+1 ≤ k 

On the other hand,

α_k ·(x ⊙y)^♭
=(α_k+1 ·(x ⊙y))^♭
=
(α_k+1 x ⊙y)^♭     if 1 ≤ k+1 ≤ i+1 

(x ⊙α_k-i y)^♭     if i+2 ≤ k+1


This proves that f is -linear. The proof for g is easier, and left to the reader.


 
Let λ=λ_1 ⋯λ_k be a constraint word with k ≥ 1, and let μ=λ_2 ⋯λ_k. Then

Σ(^λ) = 
^λ ⊕^μ     if λ_1= 

^μ     if λ_1=


The analogous result holds in the ungraded case as well.



It is clear that Σ(^1)=^0 and that Σ(^1)=^0 ⊕^1. By Proposition <ref>, we have

^λ = 
^1 ⊙^μ     if λ_1= 

^1 ⊙^μ     if λ_1=


The computation of Σ(^λ) thus follows from the previous proposition. For the ungraded case, simply note that Φ(Σ(^λ))=Σ(Φ(^λ)) as ^λ is pure.


 
The following examples follow immediately from the above propositions:


  * We have Σ(^n) = ^n ⊕^n-1 for n ≥ 1.

  * We have Σ(^n)=^n-1 for n ≥ 1.

  * We have Σ(^ab)=^ab⊕^b. .





 §.§ The transpose functor
 

Suppose M is a graded -module. We define a new graded -module, called the transpose of M, and denoted M^†, as follows. The underlying graded vector space of M^† is simply that of M; for x ∈ M, we write x^† when we regard it as an element of M^†. For x ∈ M_n, we define

α_i ·x^† = 
(α_n+1-i x)^†     if 1 ≤ i ≤ n 

x^†     if i>n 

We note that if 1 ≤ i ≤ n then α_i x^† has degree n+1, as required. We now verify that the above definition respects the fundamental relations, and thus defines an action of . To this end, suppose that x ∈ M_n and i>j. We must verify

α_i α_j x^† = α_j α_i-1 x^†.

If j>n then α_i α_j x^† = x^† and α_j α_i-1 x^† = x^†; note that i-1 ≥ j > n. Similarly, if i>n+1 then α_i-1 x^† = x^† and α_i α_j x^† = α_j x^†, since α_j x^† belongs to M_n or M_n+1, and so the identity holds. Thus suppose that 1 ≤ j ≤ n and j < i ≤ n+1. Then α_i α_j x^† = (α_n+2-iα_n+1-j x)^†. On the other hand, α_j α_i-1 x^† = (α_n+2-jα_n+2-i x)^†. Since n+2-j>n+2-i, we have α_n+2-jα_n+2-i = α_n+2-iα_n+1-j, and so the identity holds.

It is clear that transpose defines a covariant involution of the category (). We now examine how it interacts with the concatenation product.

 
Let M and N be graded -modules. Then we have a natural isomorphism

(M ⊙N)^† ≅N^† ⊙M^†.




Let ι (M ⊙ N)^†→ N^†⊙ M^† be the canonical isomorphism of graded vector spaces: explicitly, ι((x ⊙ y)^†)=y^†⊗ x^†. We verify that ι is -equivariant. Thus let x ∈ M_i and y ∈ N_j with i+j=n be given, and let k ∈ be given. We must show that

ι(α_k ·(x ⊙y)^†) = α_k ·ι((x ⊙y)^†)

This clearly holds if k>n, as then α_k acts by the identity on each side. Thus suppose k ≤ n.

First suppose that k ≤ j. Since j=n-i, we have n-k ≥ i. Thus

α_k ·(x ⊙y)^†
=(α_n+1-k ·(x ⊙y))^†
=(x ⊙(α_n+1-k-i y))^†
=(x ⊙(α_j+1-k y))^†,

and so

ι(α_k ·(x ⊙y)^†)=(α_j+1-k y)^† ⊙x^†.

On the other hand,

α_k ·ι((x ⊙y)^†)
= α_k ·(y^† ⊙x^†)
= (α_k ·y^†) ⊙x^†
=(α_j+1-k y)^† ⊙x^†.

Thus the identity holds in this case.

Now suppose k>j, so that n-k<i. We thus have

α_k ·(x ⊙y)^†
= (α_n+1-k ·(x ⊙y))^†
= ((α_n+1-k x) ⊙y)^†,

and so

ι(α_k ·(x ⊙y)^†) = y^† ⊙(α_n+1-k x)^†

On the other hand,

    α_k ·ι((x ⊙ y)^†)
       =α_k · (y^†⊙ x^†)
    =y^†⊙ (α_k-j· x^†) 
       =y^†⊙ (α_i+1-k+j x)^†
    =y^†⊙ (α_n+1-k x)^†.

Thus the identity holds in this case as well.


For a constraint word λ=λ_1 ⋯λ_r, let λ^† be the reversed word λ_r ⋯λ_1.

 
For a constraint word λ, we have (^λ)^†≅^λ^†.



This follows from Proposition <ref> and Corollary <ref>, together with the easily verified isomorphisms (^1)^†≅^1 and (^1)^†≅^1.




§ FINITE LENGTH MODULES
 



 §.§ Classification of simples


In <ref>, we introduced the simple modules ^n. We now prove that these exhaust the simple smooth -modules.

 
Any simple smooth -module is isomorphic to ^n for some n ∈.



Let M be a simple -module. Let ρ→_(M) be the homomorphism giving the action of  on M. Since M is smooth and non-zero, we cannot have ρ(α_r)=0 for all r. Let n ≥ 1 be minimal such that ρ(α_n)  0. Since α_n _≥ n = _≥ nα_n, we see that (ρ(α_n)) and (ρ(α_n)) are _≥ n-stable, and thus -stable since the α_r with r<n act by 0. Since ρ(α_n) is non-zero, we must have (ρ(α_n))=0 and (ρ(α_n))=M by simplicity. Thus α_n is a linear isomorphism.

We now claim that α_r=α_n for all r ≥ n. Suppose that we have proved this for r, and let us prove it for r+1. We have the relation α_r+1α_r=α_r^2, and so ρ(α_r+1) ρ(α_r) = ρ(α_r)^2. Since ρ(α_r)=ρ(α_n) is invertible on M, we can cancel it, and so we find ρ(α_r+1)=ρ(α_r)=ρ(α_n), as claimed.

It now follows that ρ(α_n)=𝕀. Indeed, suppose x ∈ M is non-zero. By smoothness, there exists r such that ρ(α_r) x = x. Since ρ(α_r)=0 for r<n, we must have r ≥ n. But then ρ(α_r) = ρ(α_n), and so ρ(α_n) x = x. Thus ρ(α_n) is the identity.

We have thus shown that

ρ(α_r) = 
0     if r<n 

𝕀    if r ≥ n 

Thus every linear subspace of M is a submodule, and so by simplicity, M must be one-dimensional. It is evidently isomorphic to ^n-1.


We have an analogous result in the graded case:


Any simple object of () is isomorphic to ^n for a unique n.



Let M be a simple graded -module. Let n be such that M_n  0. By simplicity, we must have M_≥ n=M and M_≥ n+1=0, since M_≥ n+1 is a proper submodule. Thus M is concentrated in degree n. Thus α_k with 1 ≤ k ≤ n act by 0 on M, while α_k with k>n acts by the identity. It follows that every subspace of M_n is an -submodule of M, and so M_n must be one-dimensional. Thus M_n ≅^n.


 
Suppose that  is algebraically closed. For n ∈∪{∞} and a ∈^×, let L_n,a be the one-dimensional -module where α_r acts by 0 for 1 ≤ r ≤ n and by a for r>n. Arguing as in the proof of Propsoition <ref>, one can show that every simple object of () is isomorphic to some L_n,a.





 §.§ Some Ext computations


We now compute the ^1 groups between simple objects in our various representation categories. We use the notations _, _, and _ for the  functor on the categories (), (), and () respectively.

 
For n,m ∈, we have

^1_(^n, ^m) = 

1     if m=n 

n     if m=n+1 

0     otherwise





Consider an extension

0 →^m →E →^n →0.

Let v, w be a basis for E with v ∈^m, and let A_r be the matrix for α_r on E with respect to this basis. Let δ_r>s be 1 if r>s and 0 otherwise. We have

A_r=[ δ_r>m   c_r;     0 δ_r>n ],

where c_r is some scalar. For any r,s ∈, we have

A_s A_r = [           δ_r>m δ_s>m c_r δ_s>m + c_s δ_r>n;                     0           δ_s>n δ_r>n ].

Suppose now that r<s, so that we have the identity

A_s A_r=A_rA_s-1.

Equating the upper right coefficients gives the equation

    c_r δ_s>m+c_s δ_r>n = c_s-1δ_r>m + c_r δ_s-1>n.

This holds for all 1 ≤ r <s. The other entries provide no information. We now proceed by cases.

.5Case 1: m<n.
Suppose r ≤ m. Taking s=m+1 in the identity (<ref>) gives c_r=0. Now suppose m<r≤ n. Then (<ref>) gives c_r=c_s-1+c_r δ_s-1>n for r<s; equivalently, we have c_r=c_s+c_r δ_s>n for r ≤ s. For s ≤ n this yields c_r=c_s, while for s>n, we obtain c_s=0. We have thus shown:

c_r = 
p     if m < r ≤ n 

0     if r≤ m or r>n


for some p ∈. Making a change of basis (replace w with w-p v), we can assume p=0, and thus c_r=0 for all r. Thus the extension is split.

.5Case 2: m>n+1.
Suppose r ≤ n. Taking s=n+2 in (<ref>) gives c_r=0. Now suppose n < r ≤ m. Then (<ref>) gives c_r δ_s>m+c_s=c_r for r<s. For s ≤ m we find c_r=c_s, while for s>m, we find c_s=0. We have thus shown:

c_r = 
p     if n < r ≤ m 

0     if r ≤ n or r>m


for some p ∈. Once again, making a change of basis (replace w with w-p v) we can assume p=0, and thus c_r=0 for all r. Thus the extension is split.

.5Case 3: m=n.
Suppose r≤ n. Taking s=n+1 in (<ref>) gives c_r=0. Now suppose n<r. Then (<ref>) gives c_r+c_s = c_s-1+ c_r for r<s; in other words, c_s=c_s-1 holds for all s>n+1. We thus have

c_r = 
p     if r>n 

0     if r ≤ n


for some p ∈. In this case, no change of basis allows us to modify p. Furthermore, for any p taking c_r as above gives a well-defined extension. We thus see that the  group is 1-dimensional, as claimed.

.5Case 4: m=n+1.
Suppose r=n+1. Then (<ref>) gives c_n+c_s=c_n, and thus c_s=0, for all n+1<s. It turns out this is all the information (<ref>) yields. Thus c_1, …, c_n+1 are arbitrary, and c_r=0 for r ≥ n+2. We can make a unique change of basis to ensure c_n+1=0. The n remaining parameters show that the  group has dimension n.


 
For n,m ∈, we have

^1_(^n, ^m) = 

n     if m=n+1 

0     otherwise





This follows from the proposition and two simple observations: first, the unique non-trivial self-extension of ^n (constructed in Case 3 ) is not smooth; and second, every extension of ^n+1 by ^n (as described in Case 4) is smooth. (Note that, the extension E considered in the proof of the proposition is smooth if and only if c_r=0 for r ≫ 0.)



In Corollary <ref>, we compute all  groups between simple objects.



It is not difficult to directly compute the ^1 groups between simple objects of (). However, we will not do this, since these groups can also be obtained by combining the Corollary <ref> with Theorem <ref> below.


 
The unique non-trivial self-extension of ^0 constructed in Case 3 of Proposition <ref> corresponds to the homomorphism ρ→_2() given by

ρ(σ) = [    1 ℓ(σ);    0    1 ],

where ℓ(σ) is the length of σ (<ref>).



Using the same analysis as in the proof of Proposition <ref>, one can compute ^1_(L_n,a, L_m,b) for all n, m, a, and b. (See Remark <ref> for the definition of L_n,a.)




 §.§ Invariants and coinvariants


For an abelian category , we let ^ (resp. ^) denote the full subcategory on finite length (resp. locally finite length) objects of .


The trivial representation ^0 is both projective and injective in ()^.



The groups ^1_(^n, ^0) and ^1_(^0, ^n) vanish for all n ≥ 0 by Corollary <ref>. Since the ^n account for all simple object of () (Proposition <ref>), a standard dévissage shows that ^1_(M, ^0) and ^1_(^0, M) vanish for all finite length objects M of (). Thus ^0 is projective and injective in ()^. The proposition now follows from general considerations. Injectivity in ()^ follows from a version of Baer's criterion (see Proposition <ref>). For projectivity, if M →^0 is a surjection with M locally finite then we can choose a finite length submodule N of M that surjects onto ^0, and a splitting of N →^0 gives a splitting for M →^0.



We will show (Corollary <ref>) that the trivial representation is in fact injective in the larger category (). However, it is clearly not projective in (), as the augmentation map ^1 →^0 is a non-split surjection.


 
The functors M ↦ M^_≥ n and M ↦ M__≥ n are exact on ()^. Moreover, the composition M^_≥ n→ M → M__≥ n is an isomorphism for M ∈()^.



We have M^=_(^0, M), which is an exact functor of M since ^0 is projective. Similarly, (M_)^* = _(M, ^0) is an exact functor of M since ^0 is injective, which implies that M_ is an exact functor of M. Since M^→ M is injective, so is the map on coinvariants. As (M^)_=M^, we thus see that M^→ M_ is injective. We similarly see that this map is surjective. We have thus proved the proposition for n=1. The statements for general n follow from the identities  M^_≥ n=Σ^n-1(M)^ and M__≥ n=Σ^n-1(M)_ (Proposition <ref>).




 §.§ The canonical grading


The most important consequence of our  calculations is perhaps the following result:

 
Every locally finite smooth -module admits a canonical grading. That is, the forgetful functor Φ()^→()^ is an isomorphism of categories.



Let M be a locally finite smooth -module. For n ≥ 0, let I_n(M)=M^_>n and let K_n(M) be the kernel of the map M → M__>n, i.e., _n+1 M. By Proposition <ref>, we have M=I_n(M) ⊕ K_n(M). Put G_n(M)=I_n(M) ∩ K_n-1(M), using the convention K_-1(M)=M.

We claim that the canonical map I_n-1(M) ⊕ G_n(M) → I_n(M) is an isomorphism, for all n ≥ 1. Since G_n(M) ⊂ K_n-1(M), we clearly have G_n(M) ∩ I_n-1(M)=0, and so the map is injective. To prove surjectivity, suppose that x ∈ I_n(M) is given. Since the map I_n-1(M) → M/K_n-1(M) is an isomorphism, there exists a unique element y ∈ I_n-1(M) having the same image in M/K_n-1(M) as x. Thus z=x-y belongs to K_n-1(M) ∩ I_n(M)=G_n(M). As x=y+z, this proves surjectivity.

It now follows by induction that the map f_n ⊕_k=0^n G_k(M) → I_n(M) is an isomorphism for any n. We claim that the map f ⊕_k ≥ 0 G_k(M) → M is an isomorphism. Since the restriction of f to ⊕_k=0^n G_k(M) is injective for all n, it follows that f is injective. To show surjectivity, suppose x ∈ M. Then, by smoothness, x ∈ I_n(M) for some n. Thus x ∈(f_n) ⊂(f), as required.

The isomorphism ⊕_k ≥ 0 G_k(M) → M endows the vector space M with a grading. We claim that this grading turns M into a graded -module. Thus suppose x ∈ G_n(M). Then x ∈ I_n(M), so if i>n then α_i x = x, as required. Now suppose 1 ≤ i ≤ n. We must show that α_i x ∈ G_n+1(M)=I_n+1(M) ∩ K_n(M). If j>n+1 then, applying the fundamental relation, we find α_j α_i x = α_i α_j-1 x=α_i x, as α_j-1 x = x; thus α_i x ∈ I_n+1(M). Since x ∈ K_n-1(M), we can write x=∑_j ≥ n (α_j-1) y_j for elements y_j ∈ M, all but finitely many of which vanish; this follows directly from the definition of coinvariants. Applying the fundamental relation, we find α_i x=∑_j ≥ n (α_j+1-1) α_i y_j, which belongs to K_n(M).

We thus see that M, equipped with the grading provided by the G_n's, is a graded -module. Denote this graded -module by G(M). If f  M → N is a morphism in ()^ then f clearly carries G_k(M) into G_k(N), and thus induces a map G(M) → G(N) in ()^. We thus see that G defines a functor

G ()^ →()^.

Moreover, it is clear that Φ∘ G=𝕀 (actual equality), since G(M) has M as its underlying vector space.

Suppose now that N is a locally finite graded -module, and put M=Φ(N). It follows from the definition of graded -module that I_n(M)=∑_0 ≤ k ≤ n N_k. We claim that K_n(M)=∑_k>n N_k. We check each containment separately.

To prove K_n(M) ⊂∑_k>n N_k, it suffices to show that (1-α_i) x ∈∑_k>n N_k for any x ∈ M and i>n, since K_n(M) is spanned by such elements. Thus let x and i be given. Let x=∑_j ≥ 0 x_j be the decomposition of x into its homogeneous pieces. Since x_j ∈ N_j, we have α_i x_j=x_j for j ≤ n. Thus (1-α_i) x = ∑_j>n (1-α_i) x_j, which belongs to ∑_k>n N_k.

We now prove K_n(M) ⊃∑_k>n N_k. Since N is locally finite, it suffices to show that for every finite length submodule N' of N we have N'_k ⊂ K_n(M) for k>n. We proceed by descending induction on k. For k ≫ 0, the statement is clear since N'_k=0. Suppose now that k>n and N'_k+1⊂ K_n(M), and let us show that N'_k ⊂ K_n(M). Thus let x ∈ N'_k be given. Then x=(1-α_k)x + α_k x. We have (1-α_k) x ∈ K_n(M) by definition, while α_k x ∈ N'_k+1⊂ K_n(M) by the inductive hypothesis. We have thus verified that K_n(M)=∑_k>n N_k.

Combining our descriptions of I_n(M) and K_n(M), we see that G_n(M)=N_n. We thus see that G ∘Φ=𝕀 (again, actual equality). Thus Φ and G provide mutually inverse isomorphisms between the categories ()^ and ()^.



The forgetful functor does not induce an equivalence between () and (). Indeed, it is not even fully faithful, as there are no non-zero maps ^n →^0, but there is a non-zero map ^n →^0, namely the augmentation map.




§ MULTIPLICITIES
 



 §.§ Definitions and simple results
 

For a smooth -module M and n ∈, we let μ_n(M) ∈∪{∞} be the multiplicity of the simple module ^n in M, as defined in <ref>. The main properties of this are summarized in the following proposition:

 
We have the following (for smooth -modules):


  * Given a short exact sequence

0 →M_1 →M_2 →M_3 →0,

we have μ_n(M_2)=μ_n(M_1)+μ_n(M_3).

  * If N is a subquotient of M then μ_n(N) ≤μ_n(M).

  * If M=⋃_i ∈ I M_i (directed union) then μ_n(M)=sup_i ∈ Iμ_n(M_i).

  * If μ_n(M)=0 for all n ∈ then M=0.




See <ref>.


For a graded -module M, we let _n(M) be the multiplicity of ^n in M. We note that this is simply the dimension of M_n. We write μ_n(M) in place of μ_n(Φ(M)).

 
Suppose that M is a locally finite graded -module. Then _n(M)=μ_n(M) for all n ≥ 0.



Using the properties of multiplicities, we can first reduce to the case where M is finite length, and then to the case where it is simple, where the statement is obvious.



The hypothesis in Proposition <ref> that M be locally finite is necessary. For example, if M=^1 then μ_0(M)=1 (Theorem <ref> below) while _0(M)=(M_0)=0.


The following simple bound will be useful in developing the theory of truncation functors in <ref>. It will be greatly improved in Proposition <ref>.

 
Let M be a smooth -module and let s ≤ r. Then μ_s(M) ≤μ_0(Σ^r M).



Suppose that μ_s(M) ≥ n. Then, by definition, we can find a chain

F^1 ⊂G^1 ⊂⋯⊂F^n ⊂G^n

of -submodules of M such that G^i/F^i ≅^s for each 1 ≤ i ≤ n. Since Σ is exact and Σ^r(^s)=^0 (Proposition <ref>, or direct observation), the chain

Σ^r(F^1) ⊂Σ^r(G^1) ⊂⋯⊂Σ^r(F^n) ⊂Σ^r(G^n)

shows that μ_0(Σ^r M) ≥ n, and so the result follows.




 §.§ The multiplicity one theorem
 

The purpose of this section is to prove the following important theorem:

 
The trivial representation has multiplicity one in any principal module; that is, we have μ_0(^r)=1 for all r ∈.



The augmentation map ϵ^r →^0= shows that μ_0(^r) ≥ 1. The content of the theorem is that (ϵ) has no trivial subquotient.


The rest of <ref> is devoted to the proof. We begin by giving a criterion for vanishing of μ_0. Let ϵ[] → be the augmentation map. We say that an element x of an -module is anti-trivial if there exists a ∈[] with ϵ(a)  0 such that ax=0.

 
Let M be an -module. Then μ_0(M)=0 if and only if every element of M is anti-trivial.



Suppose every element of M is anti-trivial. Also suppose, by way of contradiction, that M has a trivial subquotient, that is, there is an -submodule N ⊂ M and a surjection λ N →^0 of -modules. Let x ∈ N be such that λ(x)  0. Since x is anti-trivial, there exists a ∈[] with ϵ(a)  0 such that ax=0. But then 0=λ(ax)=ϵ(a) λ(x)  0, a contradiction. Thus M has no trivial subquotient, and so μ_0(M)=0.

Conversely, suppose that μ_0(M)=0 and let x ∈ M. Let I ⊂[] be the kernel of the augmentation map ϵ and let N be the submodule of M generated by x. Then  acts trivially on N/I N, and so N/IN=0 since M has no trivial subquotient. We thus find N ⊂ IN, and so x ∈ Ix. Writing x=bx for some b ∈ I, we have ax=0 with a=1-b. As ϵ(a)=1, this shows that x is anti-trivial.


 
Let K be a smooth -module with μ_0(K)=0, let N be a pure graded -module, and let M=N ⊙ K. Then μ_0(M)=0.



Let x be an element of M. We can write x=∑_i=1^n y_i ⊙ z_i where y_i ∈ N is homogeneous of degree i and z_i ∈ K. We define the degree of x to be the maximal i with z_i  0, or -∞ if x=0. We claim that, if x  0, there exists a ∈[] with ϵ(a)  0 and (ax)<(x). Granted this, it follows that every element of M is anti-trivial (repeatedly apply the claim), and so μ_0(M)=0.

Let x  0 be given of degree n, expressed as above. Since μ_0(K)=0, every element of K is anti-trivial, and so there exists b ∈[] with ϵ(b)  0 such that bz_n=0. Let f_r [] →[_>r] be the ring isomorphism mapping α_i to α_i+r, and put a=f_n(b). Note that ϵ(a)  0. By definition of the  action on the concatenation product, we have

ax = ∑_i=1^n y_i ⊙f_n-i(b) z_i

The i=n term vanishses, as f_0(b)=b, and so (ax)<(x). The claim is verified.


 
Let K be the kernel of the augmentation map ϵ^1 →. Then μ_0(K)=0.



For x=∑_n ≥ 1 c_n e_n in ^1, define the degree of x to be the maximal n such that c_n  0, or -∞ if x=0. We claim that if x ∈ K is non-zero then there exists a ∈[] with ϵ(a)  0 such that (ax)<(x). Applying the claim repeatedly, we see that any element of K is anti-trivial, and so μ_0(K)=0 by Lemma <ref>.

Let x ∈ K non-zero be given of degree n. We may as well assume the coefficient of e_n in x is 1. Since ϵ(x-e_n)=-1, it follows that x-e_n is non-zero; let its degree be n-k, with k>0. Thus x=e_n+ce_n-k+y for some non-zero scalar c and some element y ∈^1 of degree <n-k. Let σ=α_n+k-1⋯α_n and τ=α_n-1⋯α_n-k. Then

σx = e_n+k+ce_n-k+y,     τx = e_n+k+ce_n+y

and so

(c+σ-τ)x = c(c+1)e_n-k+cy.

Thus, taking a=c+σ-τ, we have ϵ(a)=c  0 and (ax) ≤ n-k < (x). This verifies the claim, and completes the proof.



Taking the exact sequence

0 →K →^1 ϵ→ ^0 →0

and applying ^r-1⊙ - yields the exact sequence

0 →^r-1 ⊙K →^r →^r-1 →0.

By Lemmas <ref> and <ref>, we have μ_0(^r-1⊙ K)=0. Thus μ_0(^r)=μ_0(^r-1). Since μ_0(^0)=1 (obvious), we find μ_0(^r)=1 for all r.




 §.§ Truncations
 

Let M be a smooth -module. For r ∈, we define τ_≥ r(M) to be _r M if r ≥ 1 and M if r ≤ 0. Here _r is the ideal of [] introduced in <ref>. By Proposition <ref>, we see that τ_≥ r(M) is an -submodule of M. We also define τ^<r(M) to be M/τ_≥ r(M). Thus we have a short exact sequence

0 →τ_≥r(M) →M →τ^<r(M) →0

of smooth -modules. For r ≥ 1, we have τ^<r(M)=M__≥ r, essentially by definition of _r, while for r ≤ 0 we have τ^<r(M)=0. We call τ_≥ r and τ^<r the truncation functors. The following proposition explains the name and notation of the truncation functors:

 
Let M be a smooth -module. Then

μ_s(τ_≥r(M)) = 
μ_s(M)     if s ≥ r 

0     if s<r 
    μ_s(τ^<r(M)) = 
0     if s ≥ r 

μ_s(M)     if s<r 




We proceed in four steps.

Step 1: μ_0(τ_≥ 1(M))=0.
We have an exact sequence

0 →τ_≥1(^r) →^r →τ^<1(^r) →0,

and τ^<1(^r) ≅^0 via the augmentation map. By Theorem <ref>, we have μ_0(^r)=1. Since μ_0(^0)=1 as well, it follows from the additivity of μ_0 that μ_0(τ_≥ 1(^r))=0. Now, let F → M be a surjection with F a sum of principal modules. It follows directly from the definition of τ_≥ 1 that the map τ_≥ 1(F) →τ_≥ 1(M) is surjective. Since μ_0(τ_≥ 1(F))=0, it follows that μ_0(τ_≥ 1(M))=0 as well.

Step 2: μ_s(τ_≥ r(M))=0 for s<r.
It follows directly from the definitions that τ_≥ 1(Σ^r-1(M))=Σ^r-1(τ_≥ r(M)). Thus

μ_s(τ_≥r(M)) ≤μ_0(Σ^r-1(τ_≥r(M)) = μ_0(τ_≥1(Σ^r-1(M)) = 0,

where the inequality comes from Proposition <ref>, and the vanishing comes from Step 1.

Step 3: μ_s(τ^<r(M))=0 for s ≥ r.
This is clear: τ^<r(M)=M__≥ r-1, and so each α_i with i ≥ r-1 acts trivially on τ^<r(M). Thus τ^<r(M) cannot contain and ^s with s ≥ r as a subquotient, since α_r-1 acts by zero on ^s.

Step 4: the remaining identities.
Since μ_s(M)=μ_s(τ_≤ r(M))+μ_s(τ^>r(M)), the remaining two identities follow from the two already established.


 
Let M be a smooth -module. If M is finitely generated, then τ^<r(M) has finite length. In general, τ^<r(M) is locally of finite length.



Suppose M is finitely generated. Write M as a quotient of a finite sum F of principal modules. We have τ^<1(^r)=(^r)_=^0 for all r, which has finite length. Thus τ^<1(F) has finite length. Since τ^<1 is right-exact, we see that τ^<1(M) is a quotient of τ^<1(F), and thus of finite length. Thus the result holds for r=1. In general, note that Σ^r-1(τ^<r(M))=τ^<1(Σ^r-1(M)), and so Σ^r-1(τ^<r(M)) has finite length. This implies that τ^<r(M) has finite length, since finite length is equivalent to the underlying vector space being finite dimensional, and Σ does not change the underlying vector space.

Now suppose M is a general smooth -module. Then M is the filtered colimit of its finitely generated submodules {M_i}. Since τ^<r is cocontinuous, we see that τ^<r(M) is the colimit of the τ^<r(M_i), each of which is finite length by the previous paragraph. Thus τ^<r(M) is locally of finite length.


 
Let M be a finitely generated smooth -module. Then μ_r(M) is finite for all r ∈.



By Proposition <ref>, we have μ_r(M)=μ_r(τ^≤ r(M)). Since τ^≤ r(M) has finite length by Proposition <ref>, the result follows.


 
The functors τ_≥ r and τ^<r are exact and cocontinuous, for all r ∈.



The description of τ^<r in terms of coinvariants shows that it is cocontinuous. We now show it is exact. Consider an exact sequence

0 →M_1 →M_2 →M_3 →0

of finitely generated smooth -modules. Applying τ^<r, we obtain an exact sequence

0 →K →τ^<r(M_1) →τ^<r(M_2) →τ^<r(M_3) →0

for some finitely generated smooth -module K. Since μ_s(τ^<r(M_1))=0 for s ≥ r (Proposition <ref>), we have μ_s(K)=0 for s ≥ r as well. Suppose now that s<r. Then μ_s(τ^<r(M_i))=μ_s(M_i) (Proposition <ref>). Combining this with the first exact sequence, we find

μ_s(τ^<r(M_2))=μ_s(τ^<r(M_1))+μ_s(τ^<r(M_3)).

From the second exact sequence, we find

μ_s(τ^<r(M_2))+μ_s(K)=μ_s(τ^<r(M_1))+μ_s(τ^<r(M_3)).

Since all the quantities appearing in these equations are finite by Proposition <ref>, it follows that μ_s(K)=0. Thus K=0 by Proposition <ref>(d), and so τ^<r(M_1) →τ^<r(M_2) is injective.

Suppose now that M → N is an injection of smooth -modules. We must show that τ^<r(M) →τ^<r(N) is injective. By the previous paragraph, this holds if M and N are finitely generated. Since τ^<r is cocontinuous, we conclude that it holds in general. Precisely, we can realize M → N as a filtered colimits of injections M_i → N_i of finitely generated smooth -modules. Since each τ^<r(M_i) →τ^<r(N_i) is injective, so is the colimit, as filtered colimits are exact.

We have thus shown that τ^<r is exact and cocontinuous. It now follows from Proposition <ref> that τ_≥ r is exact and cocontinuous.


 
Let M be a smooth -module. Then ⋂_r ≥ 0τ_≥ r(M)=0. In other words, the filtration {_r M}_r ≥ 1 of M is separated.



Let N=⋂_r ≥ 0τ_≥ r(M). Let r ∈ be given. As N ⊂τ_>r(M), we have μ_r(M)=0 by Proposition <ref>. As this holds for all r, we have N=0 by Proposition <ref>(d).



Suppose M is a smooth -module. Then τ_≥ r(τ_≥ s(M))=τ_≥max(r,s)(M). Similarly, τ^<r(τ^<s(M))=τ^<min(r,s)(M).



We have τ_≥ r(τ_≥ s(M)) ⊂τ_≥max(r,s)(M). By Proposition <ref>, the two modules have the same μ_i for all i ≥ 0. Thus they are equal (the quotient has μ_i=0 for all i, and thus vanishes). The proof of the second statement is similar.



As τ_≥ r(M)=_r M for r ≥ 1, the above proposition might suggest that _r _s=_max(r,s). In fact, this is not the case: indeed, if N is the non-trivial self-extension of the trivial representation (see Remark <ref>) then _1^2 N=0 but _1 N  0. Thus _1^2 _1. The above proposition implies, however, that _1^2 M=_1 M holds for all smooth -modules M.




 §.§ Some applications


We now give some applications of the truncation functors.

 
Any injective object of ()^ remains injective in ().



Let I be an injective object of ()^. Let M ⊂ N be finitely generated smooth -modules, and let f  M → I be an -linear map. The image of f has finite length. Thus we can find r ∈ such that μ_s((f))=0 for all s ≥ r. Since μ_s(τ_≥ r(M))=0 for all s<r (Proposition <ref>), we see that τ_≥ r(M) ⊂(f). Thus f factors through τ^<r(M). Since τ^<r is exact (Proposition <ref>), the map τ^<r(M) →τ^<r(N) is injective. As both have finite length (Proposition <ref>), we see that the map τ^<r(M) → I induced by f extends to a map τ^<r(N) → I. The composition N →τ^<r(N) → I thus extends f to N. Baer's criterion (Proposition <ref>) now shows that I is injective.


 
The trivial representation ^0 is injective in ().


We now establish some results that will be needed later.

 
Let M be a smooth -module. Then

μ_r(Σ^n M) = 
μ_0(M) + ⋯+ μ_n(M)     if r=0 

μ_r+n(M)     if r>0 




The result is clear if M is simple. It thus follows for M locally finite by the formal properties of multiplicities. We now treat the general case. Consider the short exact sequence

0 →Σ^n(τ_>r+n(M)) →Σ^n(M) →Σ^n(τ^≤r+n(M)) →0.

We have

τ^≤r(Σ^n(τ_>r+n(M)))=Σ^n(τ^≤r+n(τ_>r+n(M)))=0,

and so

μ_r(Σ^n(M)) = μ_r(Σ^n(τ^≤r+n(M))).

Since τ^≤ r+n(M) is locally finite, and μ_s(τ^≤ r+n(M))=μ_s(M) for s ≤ r+n, the result follows.


 
Let M be a smooth -module and let N be a graded -module. Then we have a natural isomorphism (N ⊙ M)_≅ N_⊗ M_. In particular, we have μ_0(N ⊙ M)=μ_0(N) μ_0(M).



Let x ∈ N_n and let y ∈ M. We have

(α_i-1) (x ⊙y) = 
((α_i-1)x) ⊙y     if i ≤ n 

x ⊙((α_i-n-1) y)     if i>n 

We thus find

_1 (N ⊙M) = (_1 N) ⊙M + N ⊙(_1 M),

and so the result follows.


 
Let M be a smooth -module. Then

μ_r(^1 ⊙M) = 
μ_0(M)     if r=0 

μ_0(M) + ⋯+ μ_r-1(M)     if r>0 




For r=0 this follows from Propsosition <ref>. Now suppose r>0. We have Σ(^1)=^1 ⊕^0. Thus, by Proposition <ref>, we have

Σ(^1 ⊙M)=(^1 ⊙M) ⊕M

It follows that we have

Σ^r(^1 ⊙M)=(^1 ⊙M) ⊕M ⊕⋯⊕Σ^r-1(M).

We thus have

    μ_r(^1 ⊙ M)
       = μ_0(Σ^r(^1 ⊙ M))-μ_0(Σ^r-1(^1 ⊙ M)) 
       = μ_0(Σ^r-1(M)) 
       = μ_0(M) + ⋯ + μ_r-1(M),

where in the first and last steps we used Proposition <ref>. This completes the proof.




§ () AS A SERRE QUOTIENT OF ()
 



 §.§ The functors Ψ and Γ
 

For a graded -module M, define Ψ(M) to be the direct limit of the directed system


M_1 [r]^α_1     M_2 [r]^α_2     M_3 [r]^α_3 [r]     ⋯

Let i ≥ 1 be given. Then we have a commutative diagram


M_i [r]^α_i [d]^α_i     M_i+1 [r]^α_i+1 [d]^α_i     M_i+2 [r]^α_i+2 [d]^α_i     ⋯

M_i+1 [r]^α_i+1     M_i+2 [r]^α_i+2 [r]     M_i+3 [r]^α_i+3     ⋯

Commutativity of the squares follows from the fundamental relation (Proposition <ref>). Both rows have direct limit equal to Ψ(M). We thus get an endomorphism of Ψ(M) that we denote by α_i. Concretely, an element x of Ψ(M) is represented by an element x∈ M_n for some n ≥ i, and α_i x is represented by α_i x∈ M_n+1. From this description of α_i, it is clear that α_j α_i = α_i α_j-1 holds on Ψ(M) for j>i. Thus Ψ(M) is naturally an -module. Moreover, it is obviously smooth: indeed, if x ∈Ψ(M) is represented by x∈ M_n then α_n x is represented by α_n x∈ M_n+1, but this also represents x, and so α_n x = x. We have thus defined a functor

Ψ() →().

Basic properties of direct limits imply that Ψ is exact and cocontinuous.

We now define a functor Γ in the opposite direction. Let M be a smooth -module. Define a graded vector space Γ(M) by Γ(M)_0=0 and Γ(M)_n = M^_≥ n for n ≥ 1. For 1 ≤ i ≤ n, we define α_i Γ(M)_n →Γ(M)_n+1 to be function induced by the restriction of α_i to Γ(M)_n; we note that it does indeed take values in Γ(M)_n+1, and that α_n is the canonical inclusion. For i>n, we define α_i on Γ(M)_n to be the identity. We leave to the reader the easy verification that these maps satisfy the appropriate relations, and thus give Γ(M) the structure of a graded -module. The definition of Γ on morphisms is clear, and so we do indeed have a functor

Γ() →().

It is clear that Γ is left exact; in fact, it is not difficult to verify that it is continuous.

We now explain the relation between the functors Ψ and Γ. For this, we introduce a piece of terminology. Let M be a graded -module. We say that x ∈ M_n, with n>0, is torsion if α_n^k x = α_n+k⋯α_n x vanishes for some k ≥ 0; we also declare all degree 0 elements to be torsion. We say that M is torsion if all of its homogeneous elements are. We let ()^ be the category of torsion modules; it is a localizing subcategory of ().

 
We have the following:


  * The functors (Ψ, Γ) are naturally an adjoint pair.

  * The co-unit ΨΓ→𝕀 is an isomorphism.

  * The kernel of Ψ is the category ()^ of torsion modules.

  * Ψ induces an equivalence ()/()^→().




(a) Let M be a graded -module. There is a natural linear map M_n →Ψ(M) which lands in the _≥ n-invariants of Ψ(M): indeed, if x is the image of x∈ M_n then α_n x is represented by α_n x, which represents x. Thus we have a map M_n →Ψ(M)^_≥ n=Γ(Ψ(M))_n, and so a map of graded vector spaces M →Γ(Ψ(M)). One easily verifies that it is -equivariant. This is the unit of the adjunction.

Now suppose that M is a smooth -module. Then we have an inclusion Γ(M)_n=M^_≥ n→ M. Moreover, the diagram

 Γ(M)_n [r]^α_n [rd]     Γ(M)_n+1 [d] 
     M 

commutes, as α_n Γ(M)_n →Γ(M)_n+1 is the canonical inclusion. Taking direct limits, we get a map Ψ(Γ(M)) → M, which is the co-unit of the adjunction. We leave to the reader the verification of the compaitiblities of the unit and co-unit.

(b) Let M be a smooth -module. Since each map Γ(M)_n → M is injective and every element of M belongs to the image of one of these maps, the map on the direct limit Ψ(Γ(M)) → M is an isomorphism.

(c) Suppose that M is a graded -module and Ψ(M)=0. This means that every element x ∈ M_n, for any n, represents 0 in the direct limit; by definition, this means that x is killed after applying some number of the transition maps, which exactly says that x is torsion.

(d) This follows from (a)–(c) and <cit.>.


 
The functor Γ takes injective objects of () to injective objects of ().




 §.§ Computing Ψ and Γ


We now prove some general results about Ψ and Γ, and use them to compute what these functors do to standard modules.

 
Let M and N be graded -modules. We then have a canonical isomorphism

Ψ(M ⊙N) ≅[ M ⊙Ψ(N_+) ] ⊕[ Ψ(M) ⊗N_0 ].




It suffices to treat separately the case where N is pure and that where it is concentrated in degree 0. The latter case is trivial, so we now treat the first. Thus suppose N is pure. By definition, Ψ(M ⊙ N) is the direct limit of the system


(M ⊙N)_1 [r]^α_1     (M ⊙N)_2 [r]^α_2 [r]     ⋯

Consider a pure tensor x ⊙ y in (M ⊙ N)_n, where x ∈ M_i and y ∈ N_j. Since N_0=0, this element vanishes if j=0. Thus suppose j>0. Then i<n, and so α_n · (x ⊙ y) = x ⊙ (α_n-i y). Thus the transition maps in the above system are the identity on the M piece. More precisely, if M_i ⊙ N denotes the evident subspace of M ⊙ N, then (M_i ⊙ N)_n is mapped into (M_i ⊙ N)_n+1 under the transition map. Furthermore, the M_i ⊙ N piece of the system is

@C=2cm
(M_i ⊙N)_n+1 [r]^α_n+1 @=[d]     (M_i ⊙N)_n+2 [r]^-α_n+2 [r] @=[d]     ⋯

M_i ⊗N_n+1-i [r]^𝕀⊗α_n+1-i     M_i ⊗N_n+2-i [r]^-𝕀⊗α_n+2+i [r]     ⋯

The direct limit of the bottom system is simply M_i ⊗Ψ(N). Since M is the sum of the M_i's, we see that Ψ(M ⊙ N) is the sum of the spaces M_i ⊗Ψ(N), which is the space underlying M ⊙Ψ(N). We thus have a natural linear isomorphism Ψ(M ⊙ N) ≅ M ⊙Ψ(N). It is easily seen to be compatible with the -actions.


 
Let λ=λ_1 ⋯λ_r be a constraint word with r ≥ 1, and let μ=λ_1 ⋯λ_r-1. Then

Ψ(^λ) ≅
^μ     if λ_r= 

0     if λ_r=





By Proposition <ref>, we have ^λ=^μ⊙^λ_r. Proposition <ref> thus yields Ψ(^λ) ≅^μ⊙Ψ(^λ_r). If λ_r= then ^λ_r=^1, and it is clear that Ψ(^1)=0; thus Ψ(^λ)=0. If λ_r= then ^λ_r=^1, and it is clear that Ψ(^1)=^0; thus Ψ(^λ) ≅^μ.


 
In the case λ_r= above, the isomorphism Ψ(^λ) ≅^μ admits a simple description: it maps the element of Ψ(^λ) represented by e_i_1,…,i_r to e_i_1,…,i_r-1.



Recall (Proposition <ref>) that for a graded -module M there is a canonical morphism M →Σ(M). Define the reduced shift of M, denoted Σ^(M), to be the cokernel of this map. Using Proposition <ref>, one can show that Σ^(^r) ≅^r-1. We thus have an isomorphism Ψ(M) ≅Φ(Σ^(M)) whenever M is a principal module. In fact, the functors Ψ and Φ∘Σ^ are not isomorphic: indeed, Ψ kills ^n, while Φ∘Σ^ does not (for n ≥ 1). There is a precise relationship between Ψ, Φ, and the shift functor (or, more precisely, its left adjoint), however: see <ref>.


 
Let M be a graded -module. Then we have a natural isomorphism Γ(Φ(M))=M ⊙^1.



We have

Γ(Φ(M))_n = Φ(M)^_≥n = ⊕_k=0^n-1 M_k,

where the second equality follows directly from the definition of graded -module. On the other hand, we have

(M ⊙^1)_n = ⊕_k=0^n-1 M_k ⊙e_n-k.

We thus have an isomorphism Γ(Φ(M))_n ≅ (M ⊙^1)_n taking x ∈ M_k to x ⊙ e_n-k. This isomorphism is easily verified to be compatible with the  actions.


 
Let λ=λ_1 ⋯λ_r be a constraint word, and let μ=λ_1 ⋯λ_r ·. Then we have a natural isomorphism Γ(^λ) ≅^μ.



We have

Γ(^λ) ≅Γ(Φ(^λ)) ≅^λ ⊙^1 ≅^μ,

where in the second step we used Proposition <ref> and in the third Proposition <ref>.


 
We have Γ(^r)=^r+1 for all r ∈.


 
The isomorphism in Proposition <ref> admits a simple direct description: it takes e_i_1,…,i_r∈Γ(^λ)_n=(^λ_n)^_≥ n to e_i_1,…,i_r,n∈^μ_n.












 §.§ Some Γ-acyclic modules


The following result gives an important source of Γ-acyclic objects.

 
Let M be a smooth -module that admits a grading (i.e., M belongs to the essential image of Φ). Then M is Γ-acyclic, that is, ^i Γ(M)=0 for i>0.



First suppose that M is locally finite; any such module admits a grading, namely the canonical grading (Theorem <ref>). Let M → J^∙ be an injective resolution of M in ()^. Each J^i is injective in () (Proposition <ref>), and so Γ(J^∙) computes Γ(M). But Γ is exact on ()^, since formation of _≥ n invariants is exact here (Proposition <ref>), and so Γ(J^∙) has no higher cohomology.

We now treat the general case. Thus suppose that M=Φ(N) for some N ∈(). Recall that N_>n is the homogeneous submodule ⊕_k>n N_k of N, and N_≤ n=N/N_>n is the quotient. Since inverse limits in () are computed degreewise, we have N= N_≤ n. Since Φ is continuous (Proposition <ref>), we have M= M_n, where M_n=Φ(N_≤ n). Clearly, M_n is locally finite.

Now, by general theory (Proposition <ref>), Γ commutes with derived limits. We thus find

    Γ( M_n) = Γ(M_n).

By the first paragraph, Γ(M_n)=Γ(M_n), since each M_n is locally finite. Moreover, since M_n+1→ M_n is surjective and Γ is exact on locally finite modules, we see that Γ(M_n+1) →Γ(M_n) is surjective. Thus for each degree d, the inverse system {Γ(M_n)_d}_n ≥ 0 of vector spaces satisfies the Mittag–Leffler condition, and so its higher derived limits vanish. We therefore see that the right side of (<ref>) is simply Γ(M_n).

Now, the higher derived functors of Γ are torsion, and thus killed by Ψ. Therefore, Ψ∘Γ is simply the identity functor. Applying Ψ to (<ref>), we thus find

M_n = Ψ(Γ(M_n)).

Thus the higher derived smooth limits of the inverse system {M_n} vanish, and so M= M_n. We therefore have

Γ(M)=Γ(M_n)=Γ(M_n),

and so ^i Γ(M)=0 for i>0.


 
We have ^i Γ(^λ)=0 for i>0, for any λ.



Let {M_i}_i ∈ be a cofiltered inverse system of smooth -modules. The argument in the proof of Proposition <ref> yields an isomorphism

M_i = Ψ(Γ(M_i)).

One can use this isomorphism to study . For example, if each M_i has level ≤ r (as defined in <ref>) then ^n Γ(M_i)=0 for n ≥ r+2 (Proposition <ref>), and so ^n  M_i=0 for n ≥ r+3.




§ COMPLETIONS
 



 §.§ The completion functor
 

Let M be a smooth -module. The truncations τ^<n(M) form an inverse system of -modules. We define the ungraded completion of M, denoted Ξ(M), to be the smooth inverse limit of this system. The -module τ^<n(M) is locally finite (Proposition <ref>), and therefore admits a canonical grading (Theorem <ref>). We define the graded completion of M, denoted Ξ(M), to be the inverse limit of this system in the category of graded -modules. Since the forgetful functor Φ is continuous (Proposition <ref>), we have Ξ(M)=Φ(Ξ(M)).

 
We have the following:


  * The functors (Ξ, Φ) are naturally an adjoint pair.

  * The functors Ξ and Ξ are exact.

  * The unit map ϵ_M  M →Ξ(M) is injective, for any smooth -module M. If M admits a grading, then ϵ_M is split.

  * The counit map η_N Ξ(Φ(N)) → N is a split surjection, for any graded -module N.




(a) Let M be a smooth -module. Since M naturally maps to each τ^<n(M), it naturally maps to the smooth inverse limit Ξ(M). This is the unit of the adjunction (Ξ, Φ).

Suppose now that N is a graded -module. Recall that N_≥ n=∑_k ≥ n N_k and N_<n=N/N_≥ n. The Jordan–Hölder constituents of N_<n are the ^k with k<n, and similarly for Φ(N_<n). Thus τ^<n(Φ(N_<n))=Φ(N_<n), and so the natural map Φ(N) →Φ(N_<n) induces a map τ^<n(Φ(N)) →Φ(N_<n). These are locally finite modules, and so this map is homogeneous with respect to the canonical gradings; note that the canonical grading on Φ(N_<n) is simply N_<n. Taking inverse limits in the graded category, and using that N= N_<n, we obtain a map Ξ(Φ(N)) → N. This is the counit of the adjoint (Ξ, Φ).

We leave to the reader the verification that the unit and co-unit thus defined satisfy the requisite relations to give an adjunction. The adjunction (Ξ, Φ) is very similar, and also left to the reader.

(b) Consider an exact sequence of smooth -modules:

0 →M_1 →M_2 →M_3 →0.

Since the truncation functor τ^<n is exact (Proposition <ref>), we get an exact sequence of inverse systems

0 →{τ^<n(M_1)} →{τ^<n(M_2)} →{τ^<n(M_3)} →0.

Since the transition maps in each of these systems is surjective, the inverse limit (in the graded category) remains exact. Thus Ξ is exact. Since Ξ=Φ∘Ξ is a composition of exact functors, it too is exact.

(c) The kernel of the map M →τ^<n(M) is τ_≥ n(M). Thus the kernel of the map M →Ξ(M) is ⋂_n ≥ 1τ_≥ n(M), which vanishes (Proposition <ref>). The claim that ϵ_M is split when M admits a grading is proven below.

(d) By the definition of adjunction, the composition


Φ(N) [r]^-ϵ_Φ(N)     Φ(Ξ(Φ(N)) [r]^-Φ(η_N)     Φ(N) 

is the identity. We thus see that ϵ_Φ(N) is a split injection, proving the remaining claim from (c). Furthermore, we see that Φ(η_N) is a split surjection, which implies that η_N is a split surjection by Proposition <ref>.


 
The forgetful functor Φ takes injective objects of () to injective objects of ().


We now prove a few simple facts about Ξ.

 
Let M be a smooth -module and let n ≥ r. Then the natural map Ξ(M)_r →τ^≤ n(M)_r on degree r pieces is an isomorphism.



Since limits are computed degreewise for graded modules, Ξ(M)_r is the inverse limit of the system {τ^≤ n(M)_r }_n ≥ 0. Consider the exact sequence

0 →τ_>r(τ^≤n(M)) →τ^≤n(M) →τ^≤r(M) →0.

By Proposition <ref>, the left term has μ_r=0, and so by Proposition <ref> it has _r=0. Thus the right map is an isomorphism on degree r pieces, and so the inverse system defining Ξ(M)_r is constant for n ≥ r. This proves the proposition.


 
Let M be a smooth -module. Then _r(Ξ(M))=μ_r(M) for all r ≥ 0.



We have

_r(Ξ(M))=_r(τ^≤r(M))=μ_r(τ^≤r(M))=μ_r(M),

where in the first step we used Proposition <ref>, in the second Proposition <ref>, and in the third Proposition <ref>.




 §.§ Computing Ξ
 

We now compute Ξ on standard modules. This will take a fair amount of work.

 
Let M be a smooth -module. We have a natural isomorphism

Ξ(^1 ⊙M) ≅[ ^1 ⊙Ξ(M) ] ⊕[ ^0 ⊗M_ ]

Moreover, if ι(M)  M →Ξ(M) denotes the canonical morphism, then the diagram


    ^1 ⊙M [rd]^ι(^1 ⊙M) [ld]_[𝕀⊙ι(M)] ⊕a 

[ ^1 ⊙Ξ(M) ] ⊕[ ^0 ⊗M_ ] @=[rr]       
Ξ(^1 ⊙M) 

commutes, where the bottom equality is the natural isomorphism and a is the map taking e_i ⊙ x to the image of x in M_.



The proof is rather lengthy, and divided into six steps:


  * Step 1: study of a filtration on ^1 ⊙ M.

  * Step 2: definition of the map f ^1 ⊙Ξ(M) →Ξ(^1 ⊙ M).

  * Step 3: definition of the map g ^0 ⊗ M_→Ξ(^1 ⊙ M).

  * Step 4: proof that f ⊕ g is surjective.

  * Step 5: proof that f ⊕ g is an isomorphism.

  * Step 6: proof that the diagram in the statement of the proposition commutes.

We use the following notation in this proof: for an element x of an -module K, we let [x]_n denote the image of x in the truncation τ^<n(M). Recall that if n>0 then τ^<n(M) is the coinvariant space M__≥ n, while if n ≤ 0 then τ^<n(M)=0. We also put N=^1 ⊙ M. 

Step 1. 
For n ≥ 1, let F^nN be the subspace of N given by ⊕_i ≥ 1( e_i ⊙τ_≥ n-i(M) ). We claim that F^n N is an -submodule of N. Indeed, suppose x ∈τ_≥ n-i (M). If j ≤ i then α_j (e_i ⊙ x)=e_i+1⊙ x, and this belongs to e_i+1⊙_n-i-1 M since τ_≥ n-i(M) ⊂τ_≥ n-i-1(M). If j>i then α_j (e_i ⊙ x)=e_i ⊙ (α_j-i x), and this belongs to e_i ⊙τ_≥ n-i(M) since τ_≥ n-i(M) is an -submodule of M. This establishes the claim.

We now examine the inverse limit of the system N/F^n N. We have

N/F^n N = ⊕_i=1^n-1 e_i ⊙τ^<n-i(M).

This module is locally finite, and thus admits a canonical grading. We have a natural map

    ^1 ⊙Ξ(M) →( ^1/∑_i ≥ n e_i ) ⊙ M__≥ n→ N/F^n N

by using the natural maps M__≥ n→ M__≥ n-i. One readily verifies that this is -linear. Moreover, it respects the gradings, as the middle group is locally finite and thus canonically graded. It is clear that the above maps are compatible with the maps N/F^n+1 N → N/F^n N. We thus get a natural map

^1 ⊙Ξ(M) →N/F^n N

of graded -modules. We claim this map is an isomorphism. It suffices to show that it is an isomorphism in each degree; thus fix d ≥ 0, and let n ≥ d. One easily verifies that the degree d piece of N/F^n N is ⊕_i=1^d e_i ⊙ (M__≥ n-i)_d-i. Similarly, the degree d piece of ^1 ⊙Ξ(M) is ⊕_i=1^d e_i ⊙Ξ(M)_d-i. The map Ξ(M)_d-i→ (M__≥ n-i)_d-i is an isomorphism by Proposition <ref>, as n-i ≥ d-i. We thus see that the degree d piece of the map (<ref>) is an isomorphism for n ≥ d, which yields the claim.

Step 2. 
Consider the map

f_n N →N__≥n,     f_n(e_i ⊙x) = [(e_i-e_n) ⊙x]_n.

We note that [e_n ⊙ x]_n is -invariant. Indeed it is invariant under α_i for i ≥ n by definition, while if i<n then

α_i [e_n ⊙x]_n=[e_n+1 ⊙x]_n=α_n [e_n ⊙x]_n=[e_n ⊙x]_n

In particular, we see that [e_m ⊙ x]_n=[e_n ⊙ x]_n for m ≥ n, as the left side is obtained from the right by applying α_n^m-n.

We claim that f_n is -linear. It is clear that e_i ⊙ x ↦ [e_i ⊙ x]_n is -linear, so it suffices to show that h  e_i ⊙ x ↦ [e_n ⊙ x]_n is -linear. Thus fix i and x, and let us show that h(α_j (e_i ⊙ x))=h(e_i ⊙ x) for all j ∈. First suppose that j ≤ i. Then

h(α_j (e_i ⊙x))=h(e_i+1 ⊙x)=[e_n ⊙x]_n=h(e_i ⊙x).

Now suppose that j>i. Then

h(α_j (e_i ⊙x))=h(e_i ⊙α_j-ix)=[e_n ⊙α_j-ix]_n=α_n+j-i [e_n ⊙x]_n=h(e_i ⊙x).

This verifies the claim that f_n is -linear.

We now claim that f_n kills F^n N. Thus we must show that f_n(e_i ⊙ x)=0 for all i ≥ 1 and x ∈τ_≥ n-i(M). If i ≥ n then

f_n(e_i ⊙x)=[(e_i-e_n) ⊙x]_n=[e_i ⊙x]_n - [e_n ⊙x]_n=0,

as explained above. Suppose i<n. It suffices to treat the case where x=(α_j-1) y for some j ≥ n-i and y ∈ M. We have

[e_i ⊙(α_j-1)y]_n=[e_i ⊙α_j y]_n-[e_i ⊙y]_n=α_i+j [e_i ⊙y]_n - [e_i ⊙y]_n=0,

as i+j ≥ n, and so α_i+j acts trivially on M__≥ n. The same reasoning shows that [e_n ⊙ (α_j-1)y]_n=0, and so f_n(e_i ⊙ (α_j-1)y)=0. The claim follows.

We thus see that f_n induces an -linear map

f_n N/F^n N →N__≥n.

We note that the domain and target are both locally finite, and thus carry a canonical grading, which is repsected by f_n. We claim that the diagram


N/F^n+1 N [d] [r]^f_n+1     N__≥n+1 [d] 

N/F^n N [r]^f_n     N__≥n 

commutes. Indeed, let e_i ⊙ x ∈ N be given. Taking its class in the top left group and mapping to the left gives [(e_i-e_n+1) ⊙ x]_n; mapping down gives [(e_i-e_n+1) ⊙ x]_n=[(e_i-e_n) ⊙ x]_n. This is exactly what we get by first mapping down and then to the right. We thus see that the f_n's form a map of inverse systems. Taking inverse limits, we thus get a map

f ^1 ⊙Ξ(M) →Ξ(^1 ⊙M)

of graded -modules.

Step 3.
Define a map

g_n M →N__≥n,     g_n(x)=[e_n ⊙x]_n.

For i ≥ 1, we have

g_n((α_i-1)x)=[e_n ⊙((α_i-1)x]_n=(α_n+i-1-1) [e_n ⊙x]_n=0.

We thus see that g_n induces a map

g_n M_ →N__≥n.

As  acts trivially on the source and the image lands in the -invariants of the target (by the computations from Step 1), we see that g_n is -linear. As both the source and target of g_n are locally finite, they carry the canonical grading, which g_n respects; in fact, the source is concentrated in degree 0 since  acts trivially on it. As [e_n+1⊙ x]_n=[e_n ⊙ x]_n, we see that the diagram

        N__≥n+1 [d] 

M_ [urr]^g_n+1 [rr]^g_n        N__≥n 

commutes. Thus the g_n induce a map

g M_ →Ξ(^1 ⊙M)

of graded -modules, where M_ is concentrated in degree 0. We could denote the domain of g by ^0 ⊗ M_ to more clearly express its structure.

Step 4.
Combining Steps 1 and 2, we have a map

f ⊕g (^1 ⊙Ξ(M)) ⊕(^0 ⊗M_) →Ξ(N).

We claim that this map is surjective. Let d ≥ 0 be given, and let us show that it is surjective in degree d. Let n ≥ d. Since the map Ξ(N)_d → (N__≥ n)_d is an isomorphism (Proposition <ref>), it suffices to show that the composition of the above map with this one is surjective in degree d. Thus, it suffices to show that the map

f_n ⊕g_n N/F^n N ⊕M_ →N__≥n

is surjective. For x ∈ M, we have

[e_i ⊙x]_n = [(e_i-e_n) ⊙x]_n + [e_n ⊙x]_n.

As the first term on the right belongs to the image of f_n, and the second to the image of g_n, the claim follows.

Step 5.
We now aim to show that f ⊕ g is an isomorphism. Since the completion and coinvariant functors commute with filtered colimits, it suffices to treat the case where M is finitely generated. As we have already shown that f ⊕ g is surjective, it thus suffices to show that the graded pieces of the domain and target have equal dimensions. For r ≥ 0, we have

_r(^1 ⊙Ξ(M))=∑_i=0^r-1 _i(Ξ(M))=∑_i=0^r-1 μ_i(M);

the first equality follows from the definition of the concatenation product, while the second follows from Proposition <ref>. Obviously, _r(^0 ⊗ M_) is M_=μ_0(M) for r=0, and vanishes otherwise. On the other hand, we have

_r(Ξ(^1 ⊙M))=μ_r(^1 ⊙M)=

μ_0(M)     if r=0 

μ_0(M) + ⋯+ μ_r-1(M)     if r>0 

by Propositions <ref> and <ref>. We thus see that the value of _r agrees on the domain and target, as required.

Step 6.
Finally, we show that the diagram in the statement of the proposition commutes. For this, it suffices to show that the diagram

     ^1 ⊙M [ld]_p ⊕a [rd]^q 

N/F^n N ⊕M_ [rr]^f_n ⊕g_n        N__≥n 

commutes, where p and q denote the canonical maps, and a is as in the statement of the proposition. Thus let e_i ⊙ x be a given element of ^1 ⊙ M. As f_n ∘ p is simply f_n, we have f_n(p(e_i ⊙ x))=[(e_i-e_n) ⊙ x]_n. Similarly, as g_n ∘ a is simply g_n, we have g_n(a(e_i ⊙ x))=[e_n ⊙ x]_n. We thus see that by mapping e_i ⊙ x under p ⊕ a follows by f_n ⊕ g_n, we get [(e_i-e_n) ⊙ x]_n + [e_n ⊙ x]_n = [e_i ⊙ x]_n, which is exactly the image of e_i ⊙ x under q. This completes the proof.


 
Let M be a smooth -module. Then we have a natural isomorphism

Ξ(^1 ⊙M) ≅^1 ⊙Ξ(M).

Moreover, the diagram


    ^1 ⊙M [ld]_i(^1 ⊙M) [rd]^𝕀⊙i(M) 

Ξ(^1 ⊙M) @=[rr]        ^1 ⊙M 

commutes, with notation as in Proposition <ref>.



Let ξ be denote a basis vector for ^1. For i ≥ 2 and x ∈ M, we have

(α_i-1) (ξ⊙x) = ξ⊙(α_i-1-1) x.

We thus see that, for n ≥ 2, we have

τ_≥n(^1 ⊙M) = ^1 ⊙τ_≥n-1(M)

as subspaces of ^1 ⊙ M. We thus have a natural isomorphism

τ^<n(^1 ⊙M) ≅^1 ⊙τ^<n-1(M)

of graded -modules. Taking inverse limits yields the claimed isomorphism. (We note that the inverse limit commutes with the operation ^1 ⊙ - since ^1 is finite dimensional.)

To prove that the diagram in the statement of the proposition commutes, it suffices to show that the diagram

     ^1 ⊙M [rd] [ld] 

τ^<n(^1 ⊙M) @=[rr]        ^1 ⊙τ^<n-1(M) 

commutes for all n ≥ 2. This is clear.


 
Let λ=μ· a^n be a constraint word, where either μ is the empty word or ends in b. Then

Ξ(^λ) ≅⊕_i=0^n ^μ·a^i.

Moreoever, the canonical injection ^λ→Ξ(^λ) is split, as is the canonical surjection Ξ(^λ) →^λ.



The computation of Ξ(^λ) follows from repeatedly applying Propositions <ref> and <ref>, together with the fact that if ν is a constraint word containing any b then (^ν)_=0. The claims about splittings follow from Proposition <ref>(c,d).


 
We have Ξ(^n) ≅⊕_0 ≤ k ≤ n^k.



Let M be a finitely generated smooth -module. Then Ξ(M) and Ξ(M) are finitely generated.



Choose a surjection F → M with F a finite sum of principal modules. Since Ξ is exact, the map Ξ(F) →Ξ(M) is surjective. Since Ξ(F) is finitely generated, it follows that Ξ(M) is as well. The argument for Ξ is identical.




 §.§ Two variants of completion


For a smooth -module M, we define the residual completion of M, denoted Ξ^(M) to be the cokernel of the natural map M →Ξ(M). For a graded -module M, we define the coresidual completion of M, denoted Ξ^(M), to be the kernel of the natural map Ξ(Φ(M)) → M. We thus have functors

Ξ^ () →(),     Ξ^ () →().

The main properties of these functors are summarized in the following proposition.

 
We have the following:


  * The functors Ξ^ and Ξ^ are exact and co-continuous.

  * Let λ=μ· a^n be a constraint word where either μ is the empty word or ends in b. Then

Ξ^(^λ) ≅⊕_i=0^n-1 ^μ·a^i,     Ξ^(^λ) ≅⊕_i=0^n-1 ^μ·a^i.


  * We have

Ξ^(^n) ≅⊕_i=0^n-1 ^i,     Ξ^(^n) ≅⊕_i=0^n-1 ^i.





(a) This follows from Proposition <ref>.

(b) This follows form Proposition <ref>.

(c) This is a special case of part (b).




 §.§ Some applications of completion


We now give some applications of the completion functor. Perhaps the most striking is:


The principal modules ^r and ^r are injective, for all r ∈.



If ^r is injective, then so is ^r+1=Γ(^r) by Corollary <ref>; similarly, if ^r is injective then so is ^r=Φ(^r) by Corollary <ref>. Since ^0 is injective (Corollary <ref>), the theorem follows.


We next use the completion functor to establish some fundamental properties of the functor Γ. For a smooth -module M, we define the generation degree of M, denoted g(M), to be the infimum of integers g such that M is generated by M^_≥ g. Note that if M=0 then g(M)=-∞, while if M is not generated by M^_≥ g for any g then g(M)=∞. Also note that g(M) ≤ g if and only if M is a quotient of a sum of principal modules ^r with r ≤ g.

 
Let M be a smooth -module with generation degree g<∞. Then Ξ^(M) has generation degree <g.



Choose a surjection F → M, where F is a sum of modules of the form ^r with r ≤ g. Then Ξ^(F) →Ξ^(M) is surjective, and the source is a sum of modules of the form ^r with r<g (Proposition <ref>). The result follows.


 
If M is a finitely generated smooth -module then ^n Γ(M) is finitely generated for all n, and vanishes for n>g(M).



We proceed by induction on g(M). If g(M)=0 then  acts trivially on M, and so Γ(M) is just (M) copies of ^1 and ^n Γ(M)=0 by Corollary <ref>. Thus the result holds. Suppose now g(M)>0. Consider the exact sequence

0 →M →Ξ(M) →Ξ^(M) →0.

Note that g(Ξ^(M))<g(M) by Lemma <ref>. Applying Γ, and appealing to Proposition <ref>, we obtain an exact sequence

0 →Γ(M) →Γ(Ξ(M)) →Γ(Ξ^(M)) →^1 Γ(M) →0

and isomorphisms

^n+1 Γ(M) ≅^n Γ(Ξ^(M))

for n ≥ 1. Since Γ(Ξ^(M)) is finitely generated by the inductive hypothesis, so is ^1 Γ(M). Similarly, since ^n Γ(Ξ^(M)) is finitely generated for n ≥ 1 and vanishes for n>g(Ξ^(M)), we see that ^n Γ(M) is finitely generated for n ≥ 2 and vanishes for n>g(M).

It remains to show that Γ(M) itself is finitely generated. For this, choose a surjection F → M, where F is a finite sum of principal modules. Since Γ∘Ξ is exact (as ^n Γ∘Ξ=0 for n ≥ 1 by Proposition <ref>), the map Γ(Ξ(F)) →Γ(Ξ(M)) is surjective. By explicit computations with principal modules (Corollaries  <ref> and <ref>), Γ(Ξ(F)) is finitely generated. Thus Γ(Ξ(M)) is finitely generated as well. As Γ(M) is a submodule of this, it too is finitely generated.


 
The functor ^n Γ commutes with filtered colimits for all n ≥ 0.



The functor (-)^_≥ r obviously commutes with filtered colimits, for any r, and so Γ commutes with filtered colimits. To prove the proposition, it thus suffices to show that that a filtered colimit of Γ-acyclic objects is Γ-acyclic (Proposition <ref>). Thus let {M_i}_i ∈ I be a filtered system of Γ-acyclic objects. Let N_i=Ξ(M_i), and let C_i=Ξ^(M_i). Since N_i=Φ(Ξ(M_i)) is Γ-acyclic (Proposition <ref>), it follows that C_i is as well. Let M, N, and C be the colimits of the systems {M_i}, {N_i}, and {C_i}. Since filtered colimits are exact, we have an exact sequence

0 →M →N →C →0

Since Ξ is cocontinuous, we have N=Ξ(M); in particular, N is Γ-acyclic. We thus have an exact sequence

0 →Γ(M) →Γ(N) →Γ(C) →^1 Γ(M) →0

and isomorphisms

^n Γ(C)=^n+1 Γ(M)

for n ≥ 1. Now, the sequence

0 →Γ(M_i) →Γ(N_i) →Γ(C_i) →0

is exact for all i, since each M_i is Γ-acyclic. Taking the direct limit, and use the fact that Γ commutes with this operation, we find that the sequence

0 →Γ(M) →Γ(N) →Γ(C) →0

is exact. Thus ^1 Γ(M)=0. Since our choice of {M_i} was arbitrary, it follows that ^1 Γ vanishes on all filtered colimits of acyclic objects. In particular, 0=Γ^1(C) ≅Γ^2(M). Continuing in this way, we find ^n Γ(M)=0 for all n, as required.


We now prove some finiteness properties of . In the graded case, we deduce the results using projective resolutions. In the smooth case, these are not available; instead, we deduce the results from the graded case by making use of properties of Γ.

 
Let M be a finitely generated graded -module and let i ∈.


  * If N is a finitely generated graded -module then ^i_(M, N) is finite dimensional.

  * The functor ^i_(M, -) commutes with filtered colimits.




Let P_∙→ M be a projective resolution of M with each P_i finitely generated. Such a resolution exists since () is locally noetherian and has enough finitely generated projectives. To prove (a), simply note that ^∙_(M, N) is computed by the complex _(P_∙, N), each term of which is finite dimensional. To prove (b), suppose that {N_i}_i ∈ is a filtered direct system in (). Then the natural map

_(P_∙, N_i) →_(P_∙, N_i)

is an isomorphism of complexes, since each term of P_∙ is a finitely generated projective. As the left complex computes ^∙_(M, N_i) and the right computes ^∙_(M,  N_i), the result follows.


 
Let M be a finitely generated smooth -module and let i ∈.


  * If N is a finitely generated smooth -module then ^i_(M, N) is finite dimensional.

  * The functor ^i_(M, -) commutes with filtered colimits.




Let M'=Γ(M), which is a finitely generated graded -module with Ψ(M') ≅ M. We have

_(M, -)=_(M', Γ(-))

by adjunction. Since both _(M', -) and Γ preserve finiteness and commute with filtered colimits (by Propositions <ref>, <ref> and <ref>), so does _(M, -).




 §.§ The right adjoint of Φ
 

The forgetful functor Φ is both continuous and cocontinuous (Proposition <ref>), and therefore has both a left and a right adjoint. As we have seen (Proposition <ref>), the left adjoint is the graded completion functor Ξ. The right adjoint, it turns out, can be described in terms of functors we have already studied. Denote the right adjoint by F. By definition, we have an isomorphism

_(M, F(N)) ≅_(Φ(M), N).

Taking M=^r, we see that F(N)_r = N^_>r. It is not difficult to see that the action of α_i on F(N)_r is simply induced by the action of α_i on N. It is now straightforward to construct an isomorphism F(N) ≅Σ(Γ(N)^†)^† of graded -modules: indeed, it follows immediately from the definitions that the two are identified as graded vector spaces, and then one simply has to work through the definitions to see that this identification is compatible with the -actions.



§ THE THEOREM ON LEVEL
 



 §.§ Level
 

Let ()_≤ r be the localizing subcategory of () generated by the standard modules ^λ of rank ≤ r. (See <ref> for background on localizing subcategories.) We define the level of a smooth -module M, denoted (M), to be the infimum of the integers r for which M belongs to ()_≤ r. By convention, the zero module has level -∞. We put ()_r=()_≤ r/()_≤ r-1, and ()_>r=()/()_≤ r. We let T_>r() →()_>r be the localization functor, and let S_>r()_>r→() be its right adjoint (the section functor).

We make similar definitions in the graded case. We let ()_≤ r be the localizing subcategory of () generated by the ^λ of rank ≤ r, and define the homogeneous level of a graded -module M, denoted (M), to be the infimum of the integers r for which M belongs to ()_≤ r. We put ()_r=()_≤ r/()_≤ r-1 and ()_>r=()/()_≤ r and define T_>r and S_>r as before.

The following proposition offers a concrete characterization of modules of level ≤ r:

 
A finitely generated smooth -module M has level ≤ r if and only if there is a finite length filtration 0=F_0 ⊂⋯⊂ F^n=M and constraint words λ_1, …, λ_n of rank ≤ r such that F^i/F^i-1 is isomorphic to a subquotient of ^λ_i for all 1 ≤ i ≤ n. The analogous statement holds in the graded case.



This follows immediately from Proposition <ref>.


We now examine how various functors interact with level.

 
We have the following:


  * Φ(()_≤ r) ⊂()_≤ r.

  * Ψ(()_≤ r) ⊂()_≤ r-1.

  * Ξ(()_≤ r) ⊂()_≤ r.

  * Ξ^(()_≤ r) ⊂()_≤ r-1.

  * Ξ^(()_≤ r) ⊂()_≤ r-1.




Since the functors in question are exact and cocontinuous, it suffices by Proposition <ref> to check the statement on the appropriate standard modules, and this follows from our explicit computations.


 
Suppose that M is a smooth -module of level ≤ r. Then ^i Γ(M) has homogeneous level ≤ r+1-i.



We first prove the result for i=0. We claim that Γ∘Ξ maps ()_≤ r into ()_≤ r+1. Since Γ∘Ξ is exact and commutes with filtered colimits, it suffices by Proposition <ref> to verify this on standard modules of level ≤ r, as these generate ()_≤ r as a localizing subcategory. This follows from explicit computations (Proposition <ref>). Now suppose that M has level ≤ r. Since natural map Γ(M) →Γ(Ξ(M)) is injective and the target has level ≤ r+1, we see that Γ(M) has level ≤ r+1 as well. This establishes the i=0 case.

We now prove the result in general by induction on level. The result is tautologically true in level <0. Suppose now that it has been proved in level <r and let us verify it in level ≤ r. Thus let M of level ≤ r be given. Consider the exact sequence

0 →M →Ξ(M) →Ξ^(M) →0.

We have (Ξ^(M))<r by Proposition <ref>. Applying Γ, and using the Γ-acyclicty of Ξ(M) (Proposition <ref>), we obtain an exact sequence

0 →Γ(M) →Γ(Ξ(M)) →Γ(Ξ^(M)) →^1 Γ(M) →0

and isomorphisms

^i+1 Γ(M) ≅^i Γ(Ξ^(M))

for i ≥ 1. By the inductive hypothesis, Γ(Ξ^(M) has level ≤ r, and so ^1 Γ(M) has level ≤ r as well. Similarly, the inductive hypothesis shows that ^i Γ(Ξ^(M)) has level ≤ r+i, and so the same is true for ^i+1Γ(M) for i ≥ 1. As we have already proved the result for i=0, we are done.


An -module (either smooth or graded) has level 0 if and only if it is locally finite. We have seen that the forgetful functor Φ()^→()^ is an equivalence. The following result generalizes this to higher level:

 
For any r ∈, the forgetful functor Φ induces an equivalence ()_r →()_r. The quasi-inverse functor is induced by Ξ.



Let M ∈()_≤ r. Then the unit M →Φ(Ξ(M))=Ξ(M) is injective (Proposition <ref>(c)) with cokernel Ξ^(M) of level <r (Proposition <ref>(d)). Thus M →Φ(Ξ(M)) is an isomorphism in ()_r.

Similarly, let N ∈()_≤ r. Then the co-unit Ξ(Φ(N)) → N is surjective (Proposition <ref>(d)) with kernel Ξ^(N) of level <r (Proposition <ref>(e)). Thus Ξ(Φ(N)) → N is an isomorphism in ()_r.




 §.§ Saturation

Fix a constraint word λ=λ_1 ⋯λ_t of rank r with λ_t=, and put μ=λ_1⋯λ_t-1. We have Ψ(^λ) ≅^μ (Proposition <ref>) and Γ(^μ) ≅^λ (Proposition <ref>). For a submodule K of ^λ, we define the saturation of K, denoted (K), to be Γ(Ψ(K)), which is naturally a submodule of ^λ.

We now make this construction somewhat more explicit. Put ^μ_≤ n=∑_k=0^n ^μ_k, regarded as a subspace of ^μ. We define the degree of a non-zero element x ∈^μ, denoted (x), to be the minimal n such that x belongs to ^μ_≤ n. Define a linear map ^μ_<n→^λ_n, denoted x ↦ x ∧ e_n, as follows:

e_i_1,…,i_t-1 ∧e_n = e_i_1,…,i_t-1,n

This map is clearly an isomorphism of vector spaces.


Let K be a homogeneous submodule of ^λ.


  * Ψ(K) consists of those elements x ∈^μ for which x ∧ e_n ∈ K for some n>(x).

  * (K) consists of all elements of the form x ∧ e_n with x ∈Ψ(K) and n>(x).




(a) Recall (Remark <ref>) the explicit form of the isomorphism Ψ(^λ) →^μ: the element of Ψ(^λ) represented by e_i_1,…,i_t∈^λ is mapped to e_i_1,…,i_t-1∈^μ. A homogeneous element of K has the form x ∧ e_n for some x ∈^μ_<n and some n. The element of Ψ(K) represented by x ∧ e_n maps to x under the isomorphism Ψ(^λ) →^μ. This proves the claim.

(b) Again, recall (Remark <ref>) the isomorphism Γ(^μ) = ^λ: the element e_i_1,…,i_t-1 of Γ(^μ)_n=^μ_<n corresponds to the element e_i_1,…,i_t-1,n=e_i_1,…,i_t-1∧ e_n of ^λ. We thus see that Γ(Ψ(K))_n consists of elements of those elements of the form x ∧ e_n with x ∈Ψ(K) ∩^μ_<n, which proves the claim.


Let N_k ⊂^λ be the homogeneous submodule spanned by those basis vectors e_i_1,…,i_t with i_t-i_t-1≥ k.


Let ν(j)=λ_1 ⋯λ_t-1·^j. Then ^λ/N_k has a finite length filtration with graded pieces ^ν(j) for 1 ≤ j ≤ k-1. In particular, ^λ/N_k has homogeneous level ≤ r-1.



Note that N_1=^λ. The quotient N_k/N_k+1 has for a basis the set _+ of those basis vectors e_i_1,…,i_t of ^λ with i_t-i_t-1=k. Moreover, we see that the set =_+ ∪{0} is stable under the action of . Thus  is a pointed -set and N_k/N_k+1≅[] is the associated monomial module. The map ^ν(k)_+ →_+ given by e_i_1,…,i_t-1,i_t,…,i_t-1+k→ e_i_1,…,i_t-1,i_t-1+k is clearly a bijection. One easily sees that the induced bijection ^ν(k)→ is an isomorphism of pointed -sets, and so N_k/N_k+1 is isomorphic to ^ν(k). The result now follows.


 
Let K ⊂^λ be a homogeneous submodule. Then there exists k such that N_k ∩(K) ⊂ K.



Let x_1, …, x_s be a Gröbner basis for Ψ(K), as defined in <ref>. For each 1 ≤ i ≤ s, let d_i=(x_i) and let n_i>d_i be an integer such that x_i ∧ e_n_i belongs to K. Let k be the maximum value of n_i-d_i for 1 ≤ i ≤ s. We claim that N_k ∩(K) ⊂ K, which will prove the proposition.

To verify the claim, let z be a given element of N_k ∩(K). Write z=y ∧ e_m where y ∈Ψ(K) and m>(y). Put δ=(y). Since z ∈ N_k, we have m-δ≥ k. Write y = ∑_i,j c_i,jσ_i,j(x_i) where the c_i,j are scalars and the σ_i,j are elements of  such that (σ_i,j x_i) ≤(y) for all i, j, which is possible by Proposition <ref>. Now, if σ'_i,j is an element of  such that σ'_i,j(n)=σ_i,j(n) for 1 ≤ n ≤ d_i then σ'_i,j x_i=σ_i,j x. Since σ_i,j(d_i) ≤δ and n_i-d_i ≤ k ≤ m-δ, we can find such σ'_i,j with σ'_i,j(n_i)=m. Relabeling, we simply assume σ_i,j(n_i)=m. We thus have σ_i,j(x_i ∧ e_n_i)=σ_i,j(x_i) ∧ e_m for all i and j, and so ∑_i,j c_i,jσ_i,j(x_i ∧ e_n_i)=y ∧ e_m=z. Thus z ∈ K, as required.


 
Let K ⊂^λ be a homogeneous submodule. Then (K)/K has homogeneous level ≤ r-1.



Let k be such that N_k ∩(K) ⊂ K. Then the map (K)/K →^λ/N_k is injective. Since the target has homogeneous level ≤ r-1, so does the source, and the result follows.




 §.§ Concatenation with ^1
 

In this section, we study the operation of concatenating with the simple module ^1. For brevity, for a smooth -module M, we put Π(M)=^1 ⊙ M. For a graded -module M, we put Π(M)=^1 ⊙ M. We also let Π^† be the conjugation of Π by the transpose functor, i.e., Π^†=Π(M^†)^†. Explicitly, we have Π^†(M)=M ⊙^1.

 
Let M be a smooth -module. Then the map

{ -submodules of M } →{ -submodules of Π(M) },     N ↦Π(N)

is a bijection. The analogous statements hold in the graded case for both Π and Π^†.



Let ξ be a generator for ^1, so that Π(M)=ξ⊗ M. An -submodule of Π(M) has the form ξ⊗ N for some subspace N of M. To prove the proposition (in the smooth case), it suffices to show that N is an -submodule of M. But this is clear: for x ∈ N and i ≥ 1, we have α_i+1(ξ⊗ x)=ξ⊗α_i x, which belongs to ξ⊗ N by assumption; thus α_i x belongs to N, as required.

It is clear that if M is a graded module then the above correspondence takes homogeneous submodules to homogeneous submodules, in each direction. Thus the analogous statement for Π holds. The one for Π^† follows simply by taking transposes.


 
For a smooth -module M we have Π(M)≤M. The analogous results hold in the graded case for both Π and Π^†.



We must show that Π maps the subcategory ()_≤ r of () into itself. Since Π is exact and cocontinuous, it suffices by Proposition <ref> to check that Π(^λ) has level ≤ r for all standard modules ^λ of level ≤ r. This follows from the explicit computation of the concatenation product on standard modules (Proposition <ref>). The proof for Π and Π^† is similar.




 §.§ The main theorem


We now come to the main theorem on level:

 
Let λ be a constraint word of rank r. Then any proper quotient of ^λ has level <r. Similarly, any proper homogeneous quotient of ^λ has homogeneous level <r.


Consider the following two statements:


  (_r) If λ is a constraint word of rank ≤ r then any proper quotient of ^λ has level <r.

  (_r) If λ is a constraint word of rank ≤ r then any proper homogeneous quotient of ^λ has homogeneous level <r.

Clearly, to prove the theorem it is enough to prove these statements for all r. We do this by induction on r. The statements (_0) and (_0) are clear, since standard modules of rank 0 are simple, in both the graded and ungraded case.


We have (_r)  (_r).



Let λ be a constraint word of rank r and let K be a non-zero submodule of ^λ. Let ϵ^λ→Ξ(^λ) and ηΞ(^λ) →^λ be the unit and counit of the adjunction between Ξ and Φ. As Φ(η) ∘ϵ is the identity, we see that η(Ξ(K)) contains K: indeed, for x ∈ K, the element η(ϵ(x))=x belongs to η(Ξ(K)). In particular, we see that η(Ξ(K)) is non-zero. We have an exact sequence

Ξ^(^λ) →Ξ(^λ/K) →^λ/η(Ξ(K)) →0

By (_r), the right module has level <r; by Proposition <ref>(e), the left module has level <r. Thus Ξ(^λ/K) has level <r. By Proposition <ref>(a), we see that Ξ(^λ/K) has level <r. Since ^λ/K injects into this module, it too has level <r, which proves (_r).



We have (_r)  (_r+1).



Let λ=λ_1 ⋯λ_t be a constraint word of rank r+1, and let μ=λ_1 ⋯λ_t-1. We must show that any proper homogeneous quotient of ^λ has homogeneous level ≤ r.

First suppose that λ_t=. Let K be a non-zero homogeneous submodule of ^λ. We show that ^λ/K has homogeneous level ≤ r. Let K'=(K) be the saturation of K. We have an exact sequence

0 →K'/K →^λ/K →^λ/K' →0.

Since K'/K has level ≤ r (Corollary <ref>), it thus suffices to prove that ^λ/K' has level ≤ r. Consider the exact sequence

0 →Ψ(K') →^μ →C →0

By (_r), the level of C is <r; note that Ψ(K') is non-zero since K'=Γ(Ψ(K')). Applying Γ to the above sequence, we find

0 →K' →^λ →Γ(C).

Since C has level <r, it follows that Γ(C) has homogeneous level ≤ r (Proposition <ref>). Since ^λ/K' is a homogeneous submodule of Γ(C), it too has homogeneous level ≤ r, and so the result follows.

Now suppose λ_t=. We have ^λ=Π^†(^μ). Suppose that K is a non-zero submodule of ^λ. Then by Proposition <ref>, we have K=Π^†(K') for some submodule K' of ^μ, necessarily non-zero. Since the functor Π^† is exact, we see that ^λ/K=Π^†(^μ/K'). Since μ is shorter than λ, we can assume by induction that ^μ/K' has homogeneous level ≤ r. Since Π^† does not increase level (Proposition <ref>), we see that ^λ/K has homogeneous level ≤ r as well, and the result follows.


The above two lemmas establish (_r) and (_r) for all r, and so the theorem follows.



 §.§ Some consequences of the main theorem


Theorem <ref> yields the following improvement of Proposition <ref>, in which the word “subquotient” is changed to “submodule.”

 
A finitely generated smooth -module M has level ≤ r if and only if there is a finite length filtration 0=F_0 ⊂⋯⊂ F^n=M and constraint words λ_1, …, λ_n of rank ≤ r such that F^i/F^i-1 is isomorphic to a submodule of ^λ_i for all 1 ≤ i ≤ n. A similar statement holds in the graded case.



Obviously if M admits such a filtration then it has level ≤ r, so let us prove the converse. Say that M is r-good if it admits a filtration 0=F_0 ⊂⋯⊂ F^n=M with each F^i/F^i-1 isomorphic to a submodule of a rank ≤ r standard module. We note that an extension of r-good modules is r-good. We must show that if M has level ≤ r then it is r-good. We proceed by induction on r. The case r=0 is clear. Suppose now that r ≥ 1 and the satement has been proved for r-1.

First, suppose that λ has rank ≤ r and M is a subquotient of ^λ, say M=A/B with B ⊂ A ⊂^λ. If B=0 then M=A is a submodule of ^λ, and obviously r-good. If B  0 then ^λ/B has level <r by Theorem <ref>, and thus the submodule M has level <r as well. Thus, by the inductive hypothesis, M is (r-1)-good, and therefore r-good as well.

Now let M be an arbitrary finitely generated module of level ≤ r. By Proposition <ref>, we have a filtration 0=F_0 ⊂⋯⊂ F^n=M and constraint words λ_1, …, λ_n of rank ≤ r such that F^i/F^i-1 is isomorphic to a subquotient of ^λ_i. By the previous paragraph, each F^i/F^i-1 is r-good. Therefore M is r-good as well. This completes the proof.


Recall that if  is a triangulated category and S is a collection of objects in  then the triangulated subcategory of  generated by S is the smallest triangulated subcategory of  containing S. The following is the most important immediate consequence of Theorem <ref>:

 
The standard modules generate ^b_(()) as a traingulated category. More precisely, the standard modules of rank ≤ r generate ^b_(()_≤ r) as a triangulated category, for all r. The analogous statements hold in the graded case.



Let _r be the triangulated subcategory of ^b_(()_≤ r) generated by the standard modules. We must show _r=^b_(()_≤ r). We proceed by induction on r. For r<0 the statement is vacuously true. Suppose now it has been proved for r-1, and let us prove it for r.

It suffices to show that M ∈_r for all M ∈()_≤ r^. Thus let such M be given. By Proposition <ref>, we have a filtration 0=F^0 ⊂⋯⊂ F^n=M and constraint words λ_1, …, λ_n of rank ≤ r such that F^i/F^i-1 is isomorphic to a subobject, say A_i, of ^λ_i. Of course, we may assume A_i is non-zero, as otherwise we could shorten the filtration. Consider the exact sequence

0 →A_i →^λ_i →^λ_i/A_i →0.

By Theorem <ref>, the quotient ^λ_i/A_i has rank <r. Thus, by the inductive hypothesis, it belongs to _r-1, and therefore to _r. Of course, ^λ_i belongs to _r by definition. Thus A_i belongs to _r as well. Since each F^i/F^i-1≅ A_i belongs to _r, we conclude that M belongs to _r.


 
Every object of ()_r is locally of finite length. Moreover, if λ has rank r then T_≥ r(^λ) is simple in this category (assuming it is non-zero), and all simple objects are isomorphic to one of this form. The analogous statements hold in the graded case.



By Theorem <ref>, any non-trivial quotient of T_≥ r(^λ) is zero, and so T_≥ r(^λ) is either zero or simple. By Propostion <ref>, we see that every finitely generated object of ()_r admits a finite filtration with graded pieces of the form T_≥ r(^λ). This shows that every finitely generated object is of finite length, and that every simple object is isomorphic to one of the form T_≥ r(^λ). The proposition follows.



Proposition <ref> offers some improvements on Proposition <ref>.




§ GROTHENDIECK GROUPS, HILBERT SERIES, AND KRULL DIMENSION
 



 §.§ Grothendieck groups
 

Let () denote the Grothendieck group of ()^. Similarly, let () denote the Grothendieck group of ()^. We let [M] denote the class of the module M in the relevant Grothendieck group. We have several homomorphisms between these groups:

    ϕ, ψ   () →()    σ   () →() 
    ξ, γ   () →()    
    (-)^†, σ   () →().

With the exception of γ, these maps are simply induced by the corresponding functors: for example, ϕ([M])=[Φ(M)]. For γ we also have to use the derived functors:

γ([M])=∑_i ≥0 (-1)^i [^i Γ(M)].

This is well-defined by Proposition <ref>. The following is our main theorem on Grothendieck groups:

 
The classes of the standard modules ^λ form an integral basis for (). Similarly, the classes of the standard modules ^λ form an integral basis for ().



The classes of the modules ^λ span () by Theorem <ref>, and similarly in the graded case. We now prove linear independence. We begin by treating the graded case. We establish linear independence by constructing sufficiently many functionals on () to distinguish the classes in question.

Let π() →() be the endomorphism given by π(x) = γ(ψ(x^†))^†. Appealing to previous computations (Propositions <ref>, <ref>, and <ref>, and Corollary <ref>), we have

π([^λ]) = 
[^λ]     if λ_1= 

0     if λ=∅ or λ_1= 

Define endomorphisms β^, β^() →() by

β^ = (σ-1) ∘π,     β^ = σ ∘(1-π).

For a non-empty constraint word λ=λ_1 ⋯λ_t, let τ(λ)=λ_2 ⋯λ_t. Appealing to Proposition <ref>, we find

β^([^λ]) = 
[^τ(λ)]     if λ_1= 

0     if λ=∅ or λ_1= 
    β^([^λ]) = 
[^τ(λ)]     if λ_1= 

0     if λ=∅ or λ_1= 

Define β^λ() →() to be the endomorphism β^λ_t∘⋯∘β^λ_1; if λ=∅ then β^λ=𝕀. For a constraint word μ=μ_1 ⋯μ_s, we have

β^λ([^μ]) = 
[^τ^t(μ)]     if s ≥ t and μ_1 ⋯μ_t = λ 

0     otherwise 

Let ρ_∅() → be the functional given by ρ_∅([M])= M_0. Note that ρ_∅([^λ])=δ_λ,∅. For a constraint word λ, define a functional ρ_λ() → by ρ_λ=ρ_∅∘β^λ. Appealing to the above computation of β^λ, we find

ρ_λ([^μ]) = δ_λ,μ.

From the existence of these functionals, we see that the classes [^λ] are linearly independent. Indeed, suppose that ∑_λ a_λ [^λ]=0 is a linear relation then. Applying ρ_λ, we find a_λ=0. As this holds for all λ, the relation is trivial.

Linear independence of the classes [^λ] in () now follows from properties of the completion functor. Precisely, by Proposition <ref>, the map ξ() →() is upper-triangular, in the sense that ξ([^λ]) is equal to [^λ] plus lower order terms (i.e., classes of the form [^μ] where μ has smaller rank than λ). Since the [^λ] are linearly independent, it follows that the [^λ] are as well.


 
The forgetful homomorphism ϕ() →() is an isomorphism.



As ϕ([^λ])=[^λ], we see that ϕ maps a basis to a basis.


Let ()_≤ r be the Grothendieck group of the category ()^_≤ r, and similarly define ()_≤ r.

 
We have the following:


  * The classes [^λ] with λ of rank ≤ r form a basis of ()_≤ r.

  * The natural map ()_≤ r→() is injective.

The analogous results hold in the graded case.



Theorem <ref> shows that the classes [^λ] with λ of rank ≤ r span ()_≤ r. Since these classes map to linear independent classes in () by Theorem <ref>, we see that they must be linearly independent in ()_≤ r and that the map is an injection.



 
Let M be a non-zero smooth -module of level r, and let [M]=∑_λ c_λ [^λ] be the expression for [M] in terms of the basis [^λ]. Then:


  * If λ has rank >r then c_λ=0.

  * If λ has rank r then c_λ≥ 0.

  * There exists some λ of rank r such that c_λ>0.

The analogous results hold in the graded case.



By Proposition <ref>, we can find a filtration 0=F^0 ⊂⋯⊂ F^n=M and constraint words λ_1, …, λ_n of rank ≤ r such that F^i/F^i-1 is isomorphic to a submodule of ^λ_i for each i. Obviously, we can assume that each F_i/F_i-1 is non-zero. Suppose λ_i has rank r. Choose an injection F^i/F^i-1→^λ_i, and let C be the cokernel. By Theorem <ref>, C has level <r, and so [C] belongs to ()_<r. We see that [F^i/F^i-1]=[^λ] modulo ()_<r. Thus

[M]=∑_(λ_i)=r [^λ_i] ()_<r

There must be some λ_i of rank r, as otherwise M would have level <r, and so the sum above is non-empty. The statements of the proposition thus follow.


 
Let M be a finitely generated smooth -module. Then M has level ≤ r if and only if [M] belongs to ()_≤ r. The analogous statement holds in the graded case.



Let M be a graded -module. Then M has homogeneous level ≤ r if and only if Φ(M) has level ≤ r.



This follows from Corollary <ref> and the fact that the isomorphism ϕ() →() maps ()_≤ r onto ()_≤ r.


 
We have the following:


  * If λ is a constraint word of rank r then T_≥ r(^λ) is a simple object of ()_r.

  * If μ is a second constraint word of rank r such that T_≥ r(^λ) is isomorphic to T_≥ r(^μ) then λ=μ.

  * The simple objects of ()_r are exactly the objects T_≥ r(^λ) with λ of rank r.

Moreover, the analogous results hold in the graded case.



(a) We have already shown (Proposition <ref>) that T_≥ r(^λ) is either zero or simple, so it suffices now to show that it is non-zero. Suppose that it is zero. It follows that ^λ belongs to the kernel of the functor T_≥ r, i.e., the category ()_<r. Thus [^λ] belongs to ()_<r⊂(). However, the former is spanned by classes of the form [^μ] with μ of rank <r, and so this would contradict Theorem <ref>.

(b) Suppose that T_≥ r(^λ) and T_≥ r(^μ) are isomorphic. We can thus find a subobjects M ⊂^λ and N ⊂^μ and a map f  M →^μ/N such that ^λ/M, N, (f), and (f) belong to ()_<r. We have the relation

[^λ]=[^μ]+[f]-[f]+[^λ/M]-[N]

in (). As the terms on the right, other than the first, belong to ()_<r, they can be expressed in terms of the classes [ν] with ν of rank <r. By Theorem <ref>, we conclude that λ=μ.

(c) This follows from what we have already proved in Proposition <ref>.


The concatenation product ⊙ on () is a monoidal structure that is exact in each argument, and preserves finite generation. It thus endows () with the structure of an associative and unital (but not necessarily commutative) ring. Moreover, the action of () on () via concatenation product endows () with the structure of a left module over the ring (). We now determine this additional structure on the Grothendieck groups:

 
We have the following:


  * Letting {a,b} denote the non-commutative polynomial ring in the variables a and b, the unique ring homomorphism f {a,b}→() satisfying f(a)=[^1] and f(b)=[^1] is an isomorphism.

  * As a ()-module, () is free of rank one with basis [^0].




(a) By Proposition <ref>, we see that f takes a word λ in a and b to [^λ]. Thus, by Theorem <ref>, f maps a basis of {a,b} to a basis of (), and is therefore an isomorphism.

(b) It suffices to show that the map g () →() defined by g(x)=x · [^0] is an isomorphism. We have g([M])=[M] · [^0]=[M ⊙^0]=[Φ(M)]. Thus g=ϕ, which we have already shown to be an isomorphism (Corollary <ref>).




 §.§ Hilbert series
 

Let M be a finitely generated graded -module. We define the Hilbert series of M by

_M(t) = ∑_n ≥0 _n(M) t^n = ∑_n ≥0 (M_n) t^n.

Now suppose that M is a finitely generated smooth -module. We define the Hilbert series of M by

_M(t) = ∑_n ≥0 μ_n(M) t^n.


 
Let M be a finitely generated smooth -module. Then _M(t)=_Ξ(M)(t).



This follows immediately from Proposition <ref>.


 
Let M and N be finitely generated graded -modules. Then

_M ⊙N(t)=_M(t) ·_N(t).




This follows immediately from the definition of the concatenation product.


 
Let λ be a constraint word with n (resp. m) a (resp. b). Then _^λ(t)=t^n+m (1-t)^-n.



The result is clear if λ= or λ=. The general case follows from these two cases and Proposition <ref> (as well as Corollary <ref>).


 
Let M be a finitely generated graded -module. Then _M(t) is a rational function of t whose denominator is a power of 1-t. Moreover, the level of M is exactly the order of the pole of _M(t) at t=1. The analogous result holds in the smooth case.



In the graded case, this follows immediately from Propositions <ref> and <ref>. The smooth case follows from the graded case and Proposition <ref>, and the observation that (M)=(Ξ(M)). (To see this, observe that (Ξ(M)) ≤(M) by Proposition <ref>(c). On the other hand,

(M) ≤(Ξ(M)) = (Φ(Ξ(M))) ≤(Ξ(M))

where the first inequality is a consequence of the injection M →Ξ(M), and the second of Proposition <ref>(a).)



Let M be a non-zero finitely generated graded -module. Then there exists a polynomial p ∈[t] such that _n(M)=p(n) for all n ≫ 0. Moreover, the degree of p is (M)-1 (using the convention that the zero polynomial has degree -1). The analogous result holds in the smooth case.




 §.§ Krull dimension


Let  be a Grothendieck abelian category. Recall that there is a notion of Krull dimension for objects in , defined as follows. Define ^≤ -1=0. Having defined ^≤ r-1, let ^≤ r be the full subcategory of  spanned by objects that are locally of finite length in the quotient /^≤ r-1. We say that an object M of  has Krull dimension r if M belongs to ^≤ r, and r is minimal with this property.

The following proposition characterizes Krull dimension for -modules:

 
Let M be a smooth -module. Then M has Krull dimension ≤ r if and only if it has level ≤ r. The analogous statement in the graded case holds as well.



Let =(), let ^≤ r be defined as in the previous paragraph, and let _≤ r=()_≤ r be the category of objects of level ≤ r. We must show ^≤ r=_≤ r for all r. We proceed by induction on r. For r=-1 the statement is vacuously true. Thus, assuming the statement for r-1, let us prove it for r. We must show that a smooth -module has level ≤ r if and only if the image of M in /_≤ r-1 is locally of finite length. If M has level ≤ r then it is locally finite in /_≤ r-1 by Proposition <ref>. Suppose now that M is locally finite in /_≤ r-1. Let N be a finitely generated submodule of M. Thus N has finite length in /_≤ r-1. Since the simple objects of /_≤ r-1 are exactly the localizations of the standard modules of rank r (Proposition <ref>), it follows that [N] is a sum of classes of such standard modules, modulo ()_≤ r-1. Thus N has level ≤ r by Corollary <ref>. Since this holds for all finitely generated submodules N of M, we see that M has level ≤ r as well.




§ INDUCTION AND COINDUCTION
 



 §.§ Coinduction
 

Let M be a -module. We define a new -module (M), called the coinduction of M, as follows. As a vector space, (M) is [M]_0 ⊕ [M]_1, where [M]_i is just a copy of M. For x ∈ M, we denote by [x]_i the corresponding element of [M]_i. The -action is defined as follows:

    α_1 [x]_0    = [α_1 x]_0 + [x]_1    α_i [x]_0    = [α_i x]_0 
    α_1 [x]_1    = 0    α_i [x]_1    = [α_i-1 x]_1

where i ≥ 2. We leave to the reader the simple verification that (M) is a well-defined -module. It is clearly smooth if M is. If M is a graded -module then we give (M) a grading as follows: for x ∈ M_n, we declare [x]_0 to have degree n and [x]_1 to have degree n+1. We leave to the reader the simple verification that this indeed defines a graded -module. We write (M) for this graded -module. We have thus defined functors

() →(),      () →().

It is clear that both are exact and cocontinuous. The rationale behind the definition of  is explained in <ref>.

Recall from <ref> that, for a smooth -module M we have defined Π(M)=^1 ⊙ M, and for a graded -module M we have defined Π(M)=^1 ⊙ M. We also write ξ for a basis element of ^1.

 
Let M be a smooth -module. Then [M]_1 is an -submodule of (M), and the map Π(M) → [M]_1 given by ξ⊙ x ↦ [x]_1 is an isomorphism of -modules. Similarly, [M]_0 is a quotient -module of (M), and the map M → [M]_0 given by x ↦ [x]_0 is an isomorphism of -modules. We thus have a canonical exact sequence

0 →Π(M) →(M) →M →0.

The analogous statement holds in the graded case.



This is clear.


 
If M is a finitely generated (resp. finite length) smooth -module then (M) is also finitely generated (resp. finite length). The analogous statement holds in the graded case.


The main interest in coinduction stems from the following proposition:

 
The functor  is naturally the right adjoint to the shift functor Σ. The analogous statement holds in the graded case.



Define ϵ M →(Σ(M)) by ϵ(x)=[(α_1 x)^♭]_0+[x^♭]_1. We verify that this is -linear. We have

ϵ(α_1 x)
=[(α_1^2 x)^♭]_0+[(α_1 x)^♭]_1.

and

α_1 ϵ(x) = [α_1 (α_1 x)^♭]_0 + [(α_1 x)^♭]_1.

These are equal, as α_1 (α_1 x)^♭=(α_2 α_1 x)^♭=(α_1^2 x)^♭. For i ≥ 2, we have

ϵ(α_i x)
=[(α_1 α_i x)^♭]_0+[(α_i x)^♭]_1
=[α_i (α_1 x)^♭]_0+[α_i-1 x^♭]_1
=α_i ϵ(x)

Thus ϵ is -linear.

Define ηΣ((M)) → M by η([x]_0^♭)=0 and η([x]_1^♭)=x. We have

η(α_i [x]_0^♭)=η( (α_i+1 [x]_0)^♭) = η([α_i+1 x]_0^♭)=0=α_i η([x]_0^♭).

and

η(α_i [x]_1^♭)=η((α_i+1 [x]_1)^♭)=η([α_i x]_1^♭)=α_i x = α_i η([x]_1^♭).

Thus η is -linear.

We leave to the reader the simple verification that ϵ and η satisfy the necessary conditions to be the unit and counit of an adjunction. We note that if M is a graded -module then ϵ and η are homogeneous, and thus give an adjunction between the graded functors.


 
The functor () →() takes injective objects to injective objects. The analogous statement holds in the graded case.


 
Let M be a graded -module and let N be a smooth -module. Then we have a natural isomorphism

(M ⊙N) ≅[ (M) ⊙N ] ⊕[ M_0 ⊗(N) ].

If N is graded then this isomorphism is homogeneous.



If M is concentrated in degree 0, the isomorphism is clear. It thus suffices to treat the case where M is pure, which we assume from now on. We have

(M ⊙N) = [M ⊙N]_0 ⊕[M ⊙N]_1,     (M) ⊙N = ([M]_0 ⊕[M]_1) ⊙N.

Define

f (M ⊙N) →(M) ⊙N,     f([x ⊙y]_ϵ)=[x]_ϵ ⊙y

for ϵ∈{0,1}. This is clearly an isomorphism of graded vector spaces, and if N is graded then it is homogeneous. We show that it is -linear.

Suppose x ∈ M_i and y ∈ M_j so that x ⊙ y has degree n=i+j. Let k ≥ 1 and ϵ∈{0,1}. We must show

    f(α_k · [x ⊙ y]_ϵ) = α_k · f([x ⊙ y]_ϵ).

We proceed in four cases.

Case 1: ϵ=0 and k=1. We have

α_1 ·[x ⊙y]_0
= [α_1 ·(x ⊙y)]_0 + [x ⊙y]_1
= [(α_1 x) ⊙y]_0 + [x ⊙y]_1.

In the second step, we used that i ≥ 1 because M is pure. On the other hand,

α_1 ·([x]_0 ⊙y)
=(α_1 ·[x]_0) ⊙y
=([α_1 x]_0+[x]_1) ⊙y,

where, again, we used i ≥ 1 in the first step. Thus (<ref>) follows.

Case 2: ϵ=0 and k ≥ 2. We have

α_k ·[x ⊙y]_0
=[α_k ·(x ⊙y)]_0
=
[(α_k x) ⊙y]_0     if k ≤ i 

[x ⊙(α_k-i y)]_0     if k ≥ i+1


On the other hand,

α_k ·([x]_0 ⊙y)
= 
(α_k [x]_0) ⊙y     if k ≤ i 

[x]_0 ⊙(α_k-i y)     if k ≥ i+1


As α_k [x]_0=[α_k x]_0, (<ref>) follows.

Case 3: ϵ=1 and k=1. We have

α_k ·[x ⊙y]_1 = 0,

and

α_k ·([x]_1 ⊙y) = (α_k [x]_1) ⊙y = 0.

Thus (<ref>) holds.

Case 4: ϵ=1 and k ≥ 2. We have

α_k ·[x ⊙y]_1
= [α_k-1 ·(x ⊙y)]_1
= 
[(α_k-1 x) ⊙y]_1     if k-1 ≤ i 

[x ⊙(α_k-1-i y)]_1     if k-1 ≥ i+1


On the other hand,

α_k ·([x]_1 ⊙y)
= 
(α_k [x]_1) ⊙y     if k ≤ i+1 

[x]_1 ⊙(α_k-i-1 y)     if k ≥ i+2.


Here we have used that [x]_1 has degree i+1. Since α_k [x]_1=[α_k-1 x]_1, we see that (<ref>) holds.




 §.§ Splitting coinduction


Let M be a smooth -module. We say that (M) splits if the exact sequence of Proposition <ref> splits, and make the analogous definition in the graded case. We note that (M) splits if and only if (Φ(M)) splits, by Proposition <ref>. Thus we will confine our investigation of splitting to the smooth case.

Let  be the monoid generated by  and an additional element β_1 subject to the relations β_1 α_1=1 and β_1 α_k = α_k-1β_1 for k ≥ 2.


Let M be a smooth -module. Then (M) splits if and only if the action of  on M can be extended to an action of .



Suppose that ϕ M →(M) is a splitting of the canonical surjection (M) → M. We thus have ϕ(x)=[x]_0+[β_1 x]_1 for some unique β_1 x ∈ M. Suppose k ≥ 2. Then

α_k ϕ(x)=[α_k x]_0+[α_k-1 β_1 x]_1

while

ϕ(α_k x)=[α_k x]_0+[β_1 α_k x]_1.

Since ϕ is -linear, the two right sides above agree, and so β_1 α_k x=α_k-1β_1 x. Similarly,

α_1 ϕ(x) = [α_1 x]_0+[x]_1,

while

ϕ(α_1 x)=[α_1 x]_0+[β_1 α_1 x]_1,

and so β_1 α_1 x=x. We have thus extended the action of  to one of . The reasoning is entirely reversible: given an action of , we define a splitting ϕ by ϕ(x)=[x]_0+[β_1 x]_1.


 
If α_1  M → M is not injective then (M) is not split.



If the action of  extends to  then α_1 must be injective since β_1 α_1 x = x for all x.


 
Let N be a smooth -module and let M=^1 ⊙ N. Then (M) is split.



We extend the action of  on M to one of  by

β_1 ·(e_i ⊙x) = 
0     if i=1 

e_i-1 ⊙x     if i ≥ 2


We now verify the requisite identities. We have

β_1 α_1 ·(e_i ⊙x) = β_1 ·(e_i+1 ⊙x) = e_i ⊙x,

and so β_1 α_1 acts as the identity, as required. Now suppose k ≥ 2. Then

α_k ·(e_i ⊙x) = 
e_i+1 ⊙x     if k ≤ i 

e_i ⊙α_k-i x     if k ≥ i+1, 

and so

β_1 α_k ·(e_i ⊙x) = 
e_i ⊙x     if k ≤ i 

0     if i=1 

e_i-1 ⊙α_k-i x     if i ≥ 2 and k ≥ i+1. 

(Note that if i=1 then k ≥ i+1, as we have assumed k ≥ 2.) On the other hand,

α_k-1 β_1 ·(e_i ⊙x) =

0     if i=1 

e_i ⊙x     if i ≥ 2 and k ≤ i 

e_i-1 ⊙α_k-i x     if i ≥ 2 and k ≥ i+1. 

We thus find that β_1 α_k=α_k-1β_1 holds on M, as required.








 §.§ Indecomposability of coinduction


We now investigate the indecomposability of coinduction. For an -module M, let M[α_1]={x ∈ M |α_1 x =0}. This is an -submodule of M: indeed, if x ∈ M[α_1] then α_1 α_k x = α_k+1α_1 x = 0, and so α_k x ∈ M[α_1]. 


For any -module M, we have (M)[α_1]=[M]_1.



We have

α_1([x]_0+[y]_1) = [α_1 x]_0+[x]_1,

and this vanishes if and only if x=0. The result follows.



Suppose that M is an indecomposable smooth -module and (M) is non-split. Then (M) is also indecomposable.



Suppose (M)=A ⊕ B for non-zero -submodules A and B. Then

M ≅(M)/(M)[α_1] = A/A[α_1] ⊕B/B[α_1],

where the first isomorphism is a consequence of the previous lemma. Since M is indecomposable, one of the terms on the right vanishes, say B/B[α_1]. Thus α_1 vanishes on B, and so B ⊂ [M]_1. It follows that [M]_1=B ⊕ (A ∩ [M]_1). However, [M]_1 ≅Π(M) is indecomposable by Proposition <ref>, and so we see that A ∩ [M]_1=0 and B=[M_1]. Thus the surjection (M) → M maps A isomorphically onto M, and is hence split, a contradiction. Thus (M) is indecomposable.


 
Suppose that M is an indecomposable smooth -module with M[α_1]  0. Then (M) is also indecomposable.



Indeed, (M) is non-split by Corollary <ref>.




 §.§ Induction
 

We have just defined and studied the coinduction functor. We now define the induction functor. We will not need to use induction in what follows, so we keep this discussion brief; it is included just for the sake of completeness.

Let M be an -module. Define a new -module, called the induction of M, and denoted (M), as follows. As a vector space, (M)=⊕_n ≥ 0{M}_n where each {M}_n is simply a copy of M. For x ∈ M, we denote by {x}_n the corresponding element of {M}_n. The action of  on the induction is given by

α_i ·{x}_n = 
{ x }_n+1     if i ≤ n+1 

{ α_i-n-1 x }_n     if i ≥ n+2 

We leave to the reader the simple verification that this is well-defined. It is clear that if M is smooth then so is (M). If M is graded then (M) admits a grading as follows: for x ∈ M_n, we let {x}_k have degree n+k. We denote this graded -module by (M). We thus have functors

() →(),      () →().

It is clear that both are exact and cocontinuous.


The functor  is naturally the left adjoint to the shift Σ. The analogous statement holds in the grade case.



The unit ϵ M →Σ((M)) is defined by ϵ(x)={x}_0^♭. The counit η(Σ(M)) → M is defined by η({x^♭}_n)=α_1^n x. We leave the remainder of the proof to the reader.



In fact, (M) is simply isomorphic to ^1 ⊙ M, via {x}_n ↦ e_n+1⊙ x.




 §.§ Explanation of definitions
 

We now explain where the formulas for induction and coinduction come from. The shift functor Σ can be described as follows: first, restrict from  to _≥ 2; then, use the isomorphism ≅_≥ 2 given by α_i ↦α_i+1 to turn the _≥ 2-module back into an -module. Thus the left and right adjoint can be described as follows: first transfer the -module to an _≥ 2-module via the isomorphism; then apply induction or coinduction from _≥ 2 to .

Thus to understand the adjoint operations, we simply need to understand induction or coinduction from _≥ 2 to . For this, we need to understand the structure of  as a left or right _≥ 2-set. From Proposition <ref>, we find

=∐_n ≥0 α_1^n _≥2,     =_≥2 ⨿_≥2 α_1.

We thus see that if M is an _≥ 2-module then the natural map

⊕_n ≥0 α_1^n ⊗M  →[] ⊗_[_≥2] M

is an isomorphism; the nth term in this decomposition corresponds to the {M}_n summand in the definition of (M). Similarly, the map

__≥2(, M) →M ⊕M,     f ↦(f(1), f(α_1)).

is an isomorphism; the two terms on the right correspond to [M]_0 and [M]_1 in (M).



§ INJECTIVE MODULES
 



 §.§ Finite length injectives
 

For n ∈_+, let ^n=^n-1(^1); also let ^0=^0. Put ^n=Φ(^n). We remind the reader that the binomial coefficient nm vanishes for m>n, and is 1 if n=m, even for negative values of n and m.

 
We have the following:


  * ^n is an indecomposable injective in () of finite length.

  * ^n is the injective envelope of ^n.

  * Every indecomposable injective in () that is locally of finite length is isomorphic to some ^n.

  * The multiplicity of ^m in ^n is n-1m-1.

The analogous results hold in the smooth case.



(a) It is clear that ^1=^1 is injective (e.g., ^0 is injective, and so (^0)=^0 ⊕^1 is injective), and obviously indecomposable. It follows from Corollary <ref> that ^n is of finite length, from Corollary <ref> that it is injective, and from Corollary <ref> that it is indecomposable.

(b) By adjunction, the isomorphism Σ^n-1(^n) →^1 gives a non-zero map ^n →^n, which is necessarily injective (since ^n is simple). Since ^n is an indecomposable injective, it is the injective envelope of any of its non-zero submodules 08Y7, and so in particular of ^n. (The cited result is stated for modules, but holds in the present situation as well.)

(c) Let J be an indecomposable injective in () that is locally of finite length. Then J contains a non-zero finite length module, and thus contains ^n for some n. As J is the injective envelope of ^n, it is thus isomorphic to ^n 08Y4. (Same comment as before.)

(d) Working in () and using Proposition <ref>, we have

([^n])=[^n]+[^n+1]

since Π(^n)=^n+1. Thus we can write

([M]) = (1+[^1]) ⊙[M]

for finite length M. Therefore,

[^n] = ^n-1([^1]) = (1+[^1])^⊙(n-1) ⊙[^1] = ∑_i=0^n-1 n-1i [^i+1],

as claimed.




 §.§ Classification of injectives
 

Let λ be a constraint word of rank r. Write

λ=^a_0(λ)     ^a_1(λ) ⋯^a_r-1     ^a_r(λ)

with a_0(λ), …, a_r(λ) ∈, and put a(λ)=(a_0(λ), …, a_r(λ)) ∈^r+1. In what follows, we partially order ^n+1 by (a_0, …, a_r) ≤ (b_0, …, b_r) if a_i ≤ b_i for all i. Furthermore, for a=(a_0, …, a_r) ∈^n+1, we let | a |=a_0+⋯+a_r.

Now, define

^λ=^a_0(λ) ⊙^1 ⊙⋯⊙^1 ⊙^a_r(λ)

and put ^λ=Φ(^λ).

 
We have the following:


  * ^λ is an indecomposable injective object of ().

  * ^λ and ^μ are isomorphic if and only if λ=μ.

  * ^λ is the injective envelope of ^λ.

  * Every indecomposable injective object of () is isomorphic to ^λ, for a unique λ.

The analogous statements hold in the smooth case.


 
Let I be an indecomposable injective graded -module. Then J_n=^n ⊙^1 ⊙ I is indecomposable injective for all n ∈.



We first treat the n=0 case. Since I is injective, so is I^†, and so is Γ(Φ(I^†))=I^†⊙^1, and so is (I^†⊙^1)^†=J_0. Suppose J_0=A ⊕ B. Then J_0^†=I^†⊙^1 = A^†⊕ B^†, and so Φ(I^†)=Ψ(A^†) ⊕Ψ(B^†). But Φ(I^†) is indecomposable by Proposition <ref>, and so Ψ(A^†)=0 or Ψ(B^†)=0, say the latter. Thus B^† is torsion. However, I^†⊙^1 is torsion-free, and so B^†=0, and thus B=0. Thus J_0 is indecomposable.

We now treat the n=1 case. Since ^1=^1, we have J_1=Π(J_0). Thus J_1 is indecomposable by Proposition <ref>. By Corollary <ref>, (J_0) is split, and so we have a decomposition (J_0) ≅ J_0 ⊕ J_1. Since (J_0) is injective (Corollary <ref>), so is the summand J_1.

We now treat the n ≥ 2 case inductively. Since ^n-1 is pure, we have

(J_n-1)
= (^n-1 ⊙^1 ⊙I)
=(^n-1) ⊙^1 ⊙I
=J_n,

where the second isomorphism comes from Proposition <ref>. Thus J_n is injective (Corollary <ref>) and indecomposable (Corollary <ref>).



(a) This follows from inductively applying the previous lemma.

(b) From Proposition <ref>(d), we see that [^λ]=[^λ]+⋯, where the remaining terms have the form [^ν] with ν of the same rank as λ and a(ν)<a(λ). Thus if ^λ is isomorphic to ^μ then [^λ]=[^μ] and so λ=μ.

(c) Let λ be given. The injections ^a_i(λ)→^a_i(λ) induces an injection ^λ→^λ. Since ^λ is indecomposable and injective, it must be the injective envelope of ^λ.

(d) Let I be an indecomposable injective of (). Let M_1 ⊂ I be a non-zero finitely generated submodule, and let M_2 ⊂ M_1 be a non-zero submodule with an injection M_2 →^λ for some λ, which exists by Proposition <ref>. Composing with the injection ^λ→^λ, we see that M_2 is a submodule of ^λ. Thus M_2 is a subobject of both I and ^λ. Since both are indecomposable, they are both injective envelopes of M_2 08Y7, and thus isomorphic 08Y4.


 
Every injective object of ()_≤ r remains injective in (). Similarly in the smooth case.



The argument used to prove Theorem <ref>(d) implies that every indecomposable injective of ()_≤ r is isomorphic to ^λ for some λ of rank ≤ r. Since ()_≤ r is locally noetherian, every injective is a sum of ^λ's, and thus remains injective in ().




 §.§ Injective resolutions


Having understood the injective objects, we now turn our attention to injective resolutions.

 
We have an injective resolution ^n → J^∙, where J^i=(^n-i)^⊕n-1n-1-i, for any n ∈.



We proceed by induction on n. The result is clear for n=0 and n=1. Suppose now that we have proved the result for n ≥ 1, and let us prove it for n+1. Let ^n → J^∙ be the given resolution. Consider the diagram


       (J^∙) [r]     J^∙ 

0 [r]     ^n+1 [r]     (^n) [r] [u]     ^n [u] [r]     0 

Here the bottom exact sequence is the canonical exact sequence for  (Proposition <ref>). The right vertical map is the given injective resolution of ^n. The middle vertical map is also an injective resolution, as  is exact and takes ^i to ^i+1. We thus obtain an injective resolution K^∙ of ^n+1 by taking the cone of the map (J^∙) → J^∙ (and shifting by one). We have

K^i = (J^i) ⊕J^i-1 = (^n-i+1)^⊕n-1n-1-i ⊕(^n-i+1)^⊕n-1n-i = (^n-i+1)^⊕nn-i,

as required.


 
Let n,m ∈. Then

^i_(^n, ^m) = ^i_(^n, ^m)=
m-1n-1     if i=m-n 

0     otherwise 




The first equality follows from Theorem <ref>, so it suffices to prove the second. Note that _(^n, ^i) is one-dimensional if n=i and vanishes otherwise, as ^n must map into the socle of ^i, which is ^i. Let ^m → J^∙ be the resolution constructed in Proposition <ref>. Then ^∙_(^n, ^m) is computed by the complex _(^n, J^∙). If n>m then all terms in the complex vanish. Otherwise, the complex has a unique non-zero term, in degree m-n, and it has dimension m-1n-1. The result follows.


 
The standard module ^λ admits an injective resolution ^λ→ I^∙ with the following properties:


  * I^0=^λ.

  * I^n=0 for n ≫ 0.

  * I^n is a finite sum of modules of the form ^μ where μ has the same rank as λ and satisfies a(μ) ≤ a(λ) and | a(λ)-a(μ)| = n.

In particular, the injective dimension of ^λ is at most | a(λ) |. The analogous statements hold in the smooth case.



Let λ=λ_1 ⋯λ_t. For each 1 ≤ i ≤ t, let ^a_i(λ)→ J^i,∙ be the injective resolution provided by Proposition <ref>. Then

^λ →J^0,∙ ⊙^1 ⊙⋯⊙^1 ⊙J^r,∙

is the sought resolution of ^λ.



In fact, the bound on injective dimension given above is slightly suboptimal. The resolution of ^a_i(λ) constructed in Proposition <ref> has length 0 if a_i(λ)=0, and length a_i(λ)-1 otherwise. We thus see that the injective dimension of ^λ is at most | a(λ) | - k where k is the number of non-zero entries in a(λ).


 
Every finitely generated smooth or graded -module has finite injective dimension, and, in fact, admits a finite length resolution by finitely generated injective modules.



Since the ^λ's generate ^b_(()) (Theorem <ref>) and have finite injective dimension (Proposition <ref>), it follows that every finitely generated graded -module has finite injective dimension.

To prove the more precise claim, it suffices to prove that every finitely generated graded -module injects into a finitely generated injective graded -module. We know this is true for standard modules (Theorem <ref>), and thus submodules of standard modules, and so it is true for all finitely generated modules by Proposition <ref>.

The same argument applies in the smooth case.


 
Let M and N be graded -modules with N finitely generated. Then ^i_(M, N)=0 for i ≫ 0. Similarly in the smooth case.




§ LOCAL COHOMOLOGY AND SATURATION
 



 §.§ Generalities
 

We now review local cohomology and saturation in general. See <cit.> for more details (though note that our notation differs from that of loc. cit.).

Let  be a Grothendieck abelian category and let  be a localizing subcategory. We assume the following condition holds:

    -13-2Injective objects of  remain injective in .Inj

We let T →/ be the localization functor and S /→ its right adjoint. We define the saturation functor → to be the composition S ∘ T. The saturation functor is left exact, and its derived functor  will play a prominent role. We say that M ∈ is (-)saturated if the natural map M →(M) is an isomorphism, and derived (-)saturated if the natural map M →(M) is an isomorphism. For M ∈, we let (M) denote the maximal subobject of M that belongs to . This defines a left exact functor →, and we again consider the derived functors , which we refer to as local cohomology.

 
For any M ∈^+(), we have a canonical exact triangle

(M) →M →(M) →
where the first two maps are the canonical ones.



See <cit.>.


 
Let I ∈ be injective. Then we have a short exact sequence

0 →(I) →I →(I) →0

where (I) and (I) are both injective. Moreover, T(I) ∈/ is injective.



See <cit.>.


 
For M ∈, the natural map (M) → M is an isomorphism (that is, (M)=M and ^i (M)=0 for i>0) and (M)=0.



Let M → I^∙ be an injective resolution in , which is an injective resolution in  by (<ref>). Thus (M) is computed by (I^∙). But this is just I^∙, since each term belongs to , and is thus quasi-isomorphic to M (via the canonical map). Similarly, (M) is compted by (I^∙), and each term vanishes.


 
Let M ∈. The following are equivalent:


  * M is saturated.

  * _(N, M)=_/(T(N), T(M)) for all N ∈.

  * ^i_(N, M)=0 for i=0,1 and all N ∈.

Moreover, if  is locally noetherian then the above are also equivalent to:



  * ^i_(N, M)=0 for i=0,1 and all N ∈^.




The equivalence of (a) and (c) is is <cit.>. The equivalence of (b) and (c) is <cit.>. Suppose now that  is locally noetherian. Obviously, (c) implies (d). Conversely, suppose that (d) holds. Let N ∈ and let {N_i}_i ∈ I be the set of finitely generated subobjects of N. Then

_(N, M)=_(N_i, M) = _(N_i, M).

Since ^i _(N_i, M)=0 for i=0,1 by our assumption, it follows that ^i _(N, M)=0 for i=0,1, and so (c) holds.


 
Let M ∈. The following are equivalent:


  * M is derived saturated.

  * ^i_(N, M)=0 for all i ≥ 0 and all N ∈.

Moreover, if  is locally noetherian then the above are also equivalent to:



  * ^i_(N, M)=0 for all i ≥ 0 and all N ∈^.




The equivalence of (a) and (b) is proved in <cit.>. The equivalence of (b) and (c) is proved similar to the equivalence of (c) and (d) in Proposition <ref>.


 
Suppose that  is locally noetherian and that for all N ∈^ the functors ^i_(N, -) commute with filtered colimits. Then:


  * A filtered colimit of saturated objects is saturated.

  * A filtered colimit of derived saturated objects is derived saturated.

  * The functor ^i and ^i commute with filtered colimits for all i.




(a) Suppose that {M_i}_i ∈ is a filtered system of saturated objects. Let N ∈^. Then for j=0,1, we have

^j_(N, M_i) = ^j_(N, M_i) = 0,

where the first isomorphism follows from the hypothesis of the proposition, and the second from Proposition <ref>. Thus M_i is saturated by Proposition <ref>.

(b) The proof is exactly the same as that of (a).

(c) We first show that  commutes with filtered colimits. Thus let {M_i}_i ∈ be a filtered system, and consider the canonical map ϕ(M_i) →( M_i). The domain of ϕ is saturated by (a), while the target of ϕ is obviously saturated. Thus to show that ϕ is an isomorphism, it suffices to that T(ϕ) is an isomorphism. This is clear: T(ϕ) is simply the identity morphism of T(M_i). (Note that T commutes with colimits and satisfies T ∘=T.)

We next show that  commutes with filtered colimits. It suffices to show that a filtered colimit of injective objects in  is -acyclic (Proposition <ref>). Thus let {I_i}_i ∈ be a filtered system of injective objects with direct limit I. For each i ∈ I, we have the exact sequence

0 →(I_i) →I_i →(I_i) →0,

and so, taking the direct limit, we have an exact sequence

0 →(I_i) →I →(I_i) →0.

Now, (I_i) belongs to  and is thus -acyclic by Proposition <ref>. Since (I_i) is saturated and injective (by Proposition <ref>), it is thus derived saturated. Hence (I_i) is derived saturated by (b), and, in particular, -acyclic. We thus see that I is -acyclic as well.

Finally, we show that  commutes with filtered colimits. Let {M_i}_i ∈ be a filtered system. Consider the diagram


(M_i) [r] [d]     M_i [r] @=[d]     (M_i) [r] [d]     

(M_i) [r]     M_i [r]     (M_i) [r]     

where the vertical maps are the natural ones and the rows come from Proposition <ref>. Since the middle and right vertical maps are isomorphisms, so is the middle one, which completes the proof.


Suppose that ' is a second localizing subcategory of  that contains  and for which ' ⊂ satisfies (<ref>). Let ' and ' be the saturation and zeroth local cohomology functors for '.


We have the following:


  * Any two of the functors {, , ', ' } commute.

  * We have ∘'=.

  * We have ∘'='.

  * We have ∘ = ' ∘' = ' ∘=0.




See <cit.>.




 §.§ The case of -modules
 

We now apply the theory of the previous section to the categories () and (). For simplicity, we will only treat the smooth case, but everything proceeds in the same manner in the graded case. The inclusion ()_≤ r⊂() satisfies (<ref>) by Corollary <ref>. Recall that

T_>r () →()_>r = ()/()_≤r

is the localization functor, and S_>r is its right adjoint. We let _>r=S_>r∘ T_>r be the saturation functor with respect to ()_≤ r, and we let _≤ r be the zeroth local cohomology functor with respect to ()_≤ r. All the results of the previous section apply in the present case, though we will not restate them. (We note in particular that the hypothesis of Proposition <ref> is met by Propositions <ref> and <ref>.)

We now compute the derived saturation and local cohomology of standard objects:

 
Let λ be a constraint word of rank r. Then

_≤s(^λ) = 
0     if s<r 

^λ     if s ≥ r 
    _>s(^λ) = 
^λ     if s<r 

0     if s ≥ r 




By Proposition <ref>, we have an injective resolution ^λ→ I^∙ where each I^n is a sum of injective objects of the form ^μ with μ of rank r. For s<r we have (^n)=0 for each n, and so (^λ)=0. For s ≥ r we have (^n)=^n for each n, and so (^λ)=^λ. The result for _>s now follows from Proposition <ref>.



Suppose that λ is a constraint word of rank r and M is a smooth -module of level <r. Then ^i_(M, ^λ)=0 for all i ≥ 0.



Theorem <ref> shows that ^λ is derived saturated with respect to ()_<r, and so the result follows from Proposition <ref>.


 
The restriction of _≥ r to ()_≤ r is exact.



If M ∈()_<r then _≥ r(M)=0 by Proposition <ref>. Suppose now that M is a non-zero submodule of ^λ with λ of rank r. Consider the exact sequence

0 →M →^λ →N →0.

Then N belongs to ()_<r by Theorem <ref>, and so _≥ r(N)=0. By Theorem <ref>, we have ^i _≥ r(^λ)=0 for i>0. We thus see that ^i _≥ r(M)=0 for i>0, that is M is _≥ r-acyclic. It now follows from Proposition <ref> that every finitely generated smooth -module of level ≤ r is _≥ r-acyclic. Since _≥ r commutes with filtered colimits by Proposition <ref>(c), we see that all smooth -modules of level ≤ r are _≥ r-acyclic, and so the corollary follows.







Consider the map f ^2 →^1 given by f(e_1,2)=e_2. Let K be the kernel and C the cokernel. Then C ≅^1, and thus has level 0. We thus see that the sequence

0 →T_>0(K) →T_>0(^2) →T_>0(^1) →0

is exact in ()_>0. Applying S_>0, and using the fact that ^2 and ^1 are derived _>0-saturated (Theorem <ref>), we obtain an exact sequence

0 →K →^2 →^1 →^1 _>0(K) →0.

We thus see that ^1 _>0(K) ≅^1. In particular, this shows that the section and saturation functors are not exact in general.




 §.§ Semistandard modules


Fix r ∈. We say that a finitely generated smooth -module M is r-semistandard if it admits a filtration 0=F^0 ⊂⋯⊂ F^n=M by -submodules such that F^i/F^i-1 is a standard module of rank r for each 1 ≤ i ≤ n. We say that a general smooth -module is r-semistandard if it is the direct union of finitely generated r-semistandard submodules. We let ()^ be the full subcategory of () spanned by the r-semistandard modules. We make analogous definitions in the graded case. Throughout this section we work in the smooth case, but all results have analogs in the graded case.


Let λ be a constraint word of rank r. Then both ^λ and ^λ are r-semistandard.


 
Let M be a smooth -module. The following are equivalent:


  * M is r-semistandard.

  * M has level ≤ r and is _≥ r-saturated.

  * M has level ≤ r and is derived _≥ r-saturated.

  * M admits an injection resolution M → I^∙ where each I^n is a sum of modules of the form ^λ with λ of rank r.




Suppose that M is finitely generated and r-semistandard. Then M is a successive extension of standard modules of rank r. As such standard modules are derived _≥ r-saturated by Theorem <ref>, we see that M is too. Now suppose that M is an arbitrary r-semistandard module. Write M= M_i where each M_i is a finitely generated r-semistandard submodule of M. Since each M_i is derived _≥ r-saturated, we see that M is as well by Proposition <ref>(b). Thus (a) implies (c).

We now claim that for any N ∈()_r, the module S_≥ r(N) is r-semistandard. Recall that the category ()_r is locally of finite length, and the simple objects have the form T_≥ r(^λ) with λ of rank r (Proposition <ref>). First suppose that N has finite length. Thus it is a successive extension of objects of the form T_≥ r(^λ) with λ of rank r. We have ^i S_≥ r(T_≥ r(^λ))=^i _≥ r(^λ), and this is ^λ for i=0 and vanishes for i>0. We thus see that S_≥ r(N) is a successive extension of modules of the form ^λ with λ of rank r, and is therefore r-semistandard. We now treat the general case. Write N= N_i (filtered colimit of subobjects) where the N_i are finite length. Since S_≥ r commutes with filtered colimits (an easy consequence of Proposition <ref>(c)), we see that S_≥ r(N)= S_≥ r(N_i) (filtered colimit of subobjects). As each S_≥ r(N_i) is r-semistandard, we see that S_≥ r(N) is r-semistandard.

Now suppose that M has level ≤ r and is _≥ r-saturated. Then M=_≥ r(M)=S_≥ r(T_≥ r(M)). As T_≥ r(M) ∈()_r, we see that M is r-semistandard by the previous paragraph. Thus (b) implies (a). It is obvious that (c) implies (b).

Again, suppose that M has level ≤ r and is derived _≥ r-saturated. Let T_≥ r(M) → J^∙ be an injective resolution in (). Then M → S_≥ r(J^∙) is exact, since M is derived saturated, and each S_≥ r(J^n) is a sum of modules of the form ^λ with λ of rank r. Thus (c) implies (d).

Finally, suppose that M → I^∙ is an injective resolution as in (d). Thus _≥ r(M) is computed by _≥ r(I^∙), but this equals I^∙ since each I^n is _≥ r-saturated. We thus see that M →_≥ r(M) is a quasi-isomorphism, and so (c) holds.


 
The category ()^ of r-semistandard objects is an abelian subcategory of () that is closed under extensions and filtered colimits. Moreover, the functors T_≥ r and S_≥ r induce mutually quasi-inverse equivalences between ()^ and ()_r.



Let f  M → N be a map of r-semistandard objects. Let K be the kernel and C the cokernel. Then K and C are both of level ≤ r. Since _≥ r is exact on ()_≤ r (Corollary <ref>), we see that the sequence

0 →_≥r(K) →_≥r(M) →_≥r(N) →_≥r(C) →0

is exact. Consider the diagram


0 [r]     K [r] [d]     M [r]^f [d]     N [r] [d]     C [r] [d]     0 

0 [r]     _≥r(K) [r]     _≥r(M) [r]     _≥r(N) [r]     _≥r(C) [r]     0 

The middle two vertical maps are isomorphisms by Propsoition <ref>. Thus the outer two vertical maps are isomorphisms as well, and so K and C are _≥ r-saturated. Since K and C are obviously of level ≤ r, we see that they are r-semistandard by Proposition <ref>. Thus ()^ is an abelian subcategory of (). It is closed under extensions and filtered colimits by Proposition <ref> and <ref>(a).

It follows from Proposition <ref> that T_≥ r carries ()^ into ()_r, and that S_≥ r carries ()_r into ()^. For M ∈()^, the natural map M → S_≥ r(T_≥ r(M))=_≥ r(M) is an isomorphism by Proposition <ref>. For N ∈()_r, the natural map T_≥ r(S_≥ r(N)) → N is an isomorphism by general properties of Serre quotients. We thus see that T_≥ r and S_≥ r are mutually quasi-inverse equivalences.



For r>0, a subquotient of an r-semistandard module is typically not r-semistandard: e.g., any proper quotient of a standard object of rank r is not r-semistandard, since it has rank <r by Theorem <ref>. Thus ()^ is not a localizing subcategory of ().



The forgetful functor Φ induces an equivalence ()^→()^.



It is clear that Φ maps ()^ into ()^. Moreover, the diagram


()^ [r]^Φ [d]_T_≥r    
()^ [d]^T_≥r 

()_r [r]^Φ    
()_r 

commutes, up to isomorphism. The vertical maps are equivalences by Theorem <ref>, while the bottom map is an equivalence by Proposition <ref>. Thus the top map is an equivalence, which yields the result.




 §.§ Semi-orthogonal decompositions
 

Given a triangulated category  and triangulated subcategories _0, _1, …, we say that the 's give a semi-orthogonal decomposition of  if the following two conditions hold:


  * For X ∈_i and Y ∈ T_j with i<j, we have _(X,Y)=0.

  * The _i generate  as a triangulated category.

In this case, we write =⟨_0, _1, …⟩.

 
Let =^b_(()). For r ≥ 0, let _r be the full subcategory of  spanned by objects M such that ^i(M) ∈()^ for all i. Then:


  * _r is a triangulated subcategory of .

  * We have an equivalence of triangulated categories _r ≅^b_(()^).

  * We have a semi-orthogonal decomposition =⟨_0, _1, …⟩.

The analogous result holds in the graded case.



(a) This follows from the fact that ()^ is an abelian subcategory of () that is closed under extensions (Theorem <ref>).

(b) The inclusion ()^→() induces a functor F ^b_(()^) →_r. We first claim that F is fully faithful. To see this, suppose that M and N are objects in ^b_(()^). We can then find quasi-isomorphisms M → I and N → J where I^n and J^n are sums of modules of the form ^λ with λ of rank r. We thus see that (M, N), in either ^b_(()^) or in ^b_(()), is computed by _(I, J), where  is the homotopy category of complexes, and so F is fully faithful.

We now show that F is essentially surjective. Let M ∈_r be given. By Proposition <ref>(d), and basic homological algebra, we can find a quasi-isomorphism M → I where each I^n is a sum of modules of the form ^λ with λ of rank r. Thus I defines an object of ^b_(()^) and we have an isomorphism F(I) ≅ M. Thus F is essentially surjective.

(c) Suppose that M and N are -modules that are r- and s-semistandard, respectively, with r<s. By Proposition <ref>(c), N is derived _≥ s-saturated, and so the natural map

_(M, N) →_()_≥s(T_≥s(M), T_≥s(N))

is an isomorphism. Since T_≥ s(M)=0, we thus find that _(M, N)=0. It follows that _(X, Y)=0 for X ∈_r and Y ∈_s. Since the 's generate  (by Theorem <ref>, for example), the result follows.




§ STRUCTURE OF LEVEL CATEGORIES
 



 §.§ Multigraded ^r-modules


Let r ∈, and let ^r denote the r-fold direct product ×⋯×. An ^r-grading on a vector space M is a decomposition M=⊕_n∈^r M_n. A graded ^r-module is a vector space M equipped with a ^r-grading such that: (1) for x ∈ M_n and σ∈^r we have σ(x) ∈ M_σn; and (2) if x ∈ M_n and σ∈^r fixes n then σ(x)=x. We let (^r) denote the category of graded ^r-modules.

If M_1, …, M_r are graded -modules then their tensor product is naturally a graded ^r-module, which we denote by M_1 ⊠⋯⊠ M_r. This defines a functor

⊠()^r →(^r).

We say that an object of (^r) is factorizable if it belongs to the essential image of ⊠.

 
Let M_1, …, M_r and N_1, …, N_r be finitely generated graded -modules, and put M=M_1 ⊠⋯⊠ M_r and N=N_1 ⊠⋯⊠ N_r. Then the natural map

⊗_i=1^r _(M_i, N_i) →_^r(M, N)

is an isomorphism.



Suppose that we have an exact sequence

M_1” →M_1' →M_1 →0

of finitely generated graded -modules. Let M' and M” be defined like M, but using M_1' and M_1”, and let H=⊗_i=2^r _(M_i, N_i). We then get a commutative diagram


0 [r]     _(M_1, N_1) ⊗H [r] [d]     _(M_1', N_1) ⊗H [r] [d]     _(M_1”, N_1) ⊗H [d] 

0 [r]     _^r(M, N) [r]     _^r(M', N) [r]     _^r(M”, N) 

We thus see that if the right two vertical maps are isomorphisms then so is the leftmost vertical maps. Thus to prove the proposition for (M_1, …, M_r) (and all N's) it suffices to prove it for (M_1', M_2, …, M_r) and (M_1”, M_2, …, M_r). Obviously, the same holds for the other M_i's.

We can thus reduce to the case where each M_i is a finitely generated projective module. Moreover, since everything is additive, we can assume that each M_i is an indecomposable projective, say M_i=^m_i. Thus _(M_i, N_i) is (N_i)_m_i, the degree m_i piece of N_i. One easily sees that M is a projective object in (^r), and that for any graded ^r-module K we have _^r(M, K)=K_m_1,…,m_r. We thus see that

_^r(M, N)=N_m_1,…,m_r=(N_1)_m_1 ⊗⋯⊗(N_r)_m_r.

Thus the source and target of the map in question are each identified with the same space, and one easily sees that the map in question is identified with the identity map, which proves the proposition.


The functor ⊠ realizes (^r) as the r-fold tensor power of the category (). The precise meaning of this statement is exactly the following proposition:

 
Let  be a Grothendieck abelian category, and let _0 ()^r → be a functor that is cocontinuous in each variable. Then there exists a unique (up to isomorphism) functor (^r) → that is cocontinuous and for which there is a functorial isomorhpism (M_1 ⊠⋯⊠ M_r) ≅_0(M_1, …, M_r).



For (n_1, …, n_r) ∈^r, let ^n_1,…,n_r be the graded ^r-module ^n_1⊠⋯⊠^n_r. Let ⊂(^r) be the full subcategory spanned by these modules. Similarly, let _0 ⊂()^r be the full subcategory on objects of the form (^n_1, …, ^n_r). Now, consider the following functor categories:


  * _1 is the category of all cocontinuous functors (^r) →.

  * _2 is the category of all additive functors →.

  * _3 is the category of all functor ()^r → that are cocontinuous in each variable.

  * _4 is the category of all functors _0 → that are additive in each variable.

Consider the diagram of restriction functors:


_1 [r] [d]     _2 [d] 

_3 [r]     _4 

The top functor is an equivalence by Proposition <ref>. A similar argument shows that the bottom functor is also an equivalence.

We now claim that the right functor is an equivalence. Indeed, suppose that _0 ∈_4. We define a functor → as follows. First, we put (^n_1,…,n_r)=_0(^n_1,…,^n_r). Now, consider the diagram


∏_i=1^n _(^n_i, ^m_i) [r] [d]    
⊗_i=1^n _(^n_i, ^m_i) @..>[d] 

_((^n_1, …, ^n_r), (^m_1, …, ^m_r)) @=[r]    
_('(^n_1,…,n_r), '(^m_1,…,m_r)) 

The left vertical map here is the one induced by _0. Since _0 is additive in each variable, this map is additive in each variable. The dotted arrow therefore fills in uniquely as an additive map. Since the top right group is _^r(^n_1,…,n_r,^m_1,…,m_r) by Proposition <ref>, the right map defines  on morphisms. Thus  is a well-defined additive functor. One easily sees that _0 ↦ defines a functor _4 →_2 quasi-inverse to the restriction functor, which establishes the claim.

It now follows that the restriction functor _1 →_3 is an equivalence, which completes the proof.




 §.§ The functor Ω


Let Ω_0 ()^r+1→() be the functor given by

Ω_0(M_0, …, M_r) = M_0 ⊙^1 ⊙⋯⊙^1 ⊙M_r.

This is cocontinuous in each argument. Thus, by Proposition <ref>, it uniquely extends to a cocontinuous functor Ω(^r+1) →().


The functor Ω is exact.



The vector space underlying Ω_0(M_0, …, M_r) is canonically isomorphic to M_0 ⊗⋯⊗ M_r ⊗^r, where here we are simply regarding all the objects as vector spaces. Thus, if F () → denotes the forgetful functor, then F ∘Ω and M ↦ M ⊗^r are two cocontinuous functors (^r+1) → that have isomorphic restrictions to ()^r+1 via ⊠. Thus, by the uniqueness of Proposition <ref>, we see that they are isomorphic. In other words, the vector space underlying Ω(M) is canonically isomorphic to M ⊗^r. As this is exact in M, we see that Ω is exact.


 
Let M_0, …, M_r and N_0, …, N_r be graded -modules of finite length. Put M = M_0 ⊠…⊠ M_r and similarly define N. Then the canonical map

⊗_i=0^r _(M_i, N_i) →_(Ω(M), Ω(N))

is an isomorphism.



Suppose that

0 →N_0 →N_0' →N_0” →0

is an exact sequence in ()^, and put

N'=N_0' ⊠N_1 ⊠⋯⊠N_r,     N”= N_0” ⊠N_1 ⊠⋯⊠N_r.

Since ⊠ is exact in each variable and Ω is exact, we have an exact sequence

0 →Ω(N) →Ω(N') →Ω(N”) →0.

Letting H=⊗_i=1^r _(M_i, N_i), we thus have a morphism of triangles


_(M_0, N_0) ⊗H [r] [d]    
_(M_0, N'_0) ⊗H [r] [d]    
_(M_0, N'_0) ⊗H [r] [d]    


_(Ω(M), Ω(N)) [r]    
_(Ω(M), Ω(N')) [r]    
_(Ω(M), Ω(N”)) [r]     

If two of the three vertical maps are isomorphisms, then so is the third. We thus see that if the lemma holds for two of N_0, N_0', and N_0” (with the other modules fixed) then it holds for the third as well. Of course, the same holds for the other N_i and the M's.

Using the above observation, we can reduce to the case where each M_i is simple and each N_i is injective; say M_i=^a_i and N_i=^b_i. Thus if λ and μ are the constraint words with a(λ)=(a_0,…,a_r) and a(μ)=(b_0,…,b_r) then Ω(M)=^λ and Ω(N)=^μ. Since N and the N_i are injective, all the 's are concentrated in degree 0, and so it suffices to show that the map

⊗_i=0^r _(M_i, N_i) →_(Ω(M), Ω(N))

is an isomorphism. If λ=μ then all these  spaces are one-dimensional, and the map is easily seen to be non-zero, and thus an isomorphism. Now suppose λμ. We claim that both sides vanish. Indeed, we have a_i  b_i for some i, and so _(M_i, N_i)=0, as the socle of N_i is ^b_i, which is a simple object not isomorphic to M_i. Similarly, we have an injection ^μ→^μ, and so T_≥ r(^μ) is the socle of T_≥ r(^μ) in ()_r. Thus (T_≥ r(^λ), T_≥ r(^μ))=0, which implies that _(^λ, ^μ)=0 since ^μ is _≥ r-saturated.


 
Notation as in Proposition <ref>, the map

_^r+1(M, N) →_(Ω(M), Ω(N))

induced by Ω is an isomorphism. In other words, Ω is fully faithful on objects of (^r+1) that are factorizable and of finite length.



Applying ^0 to the isomorphism in Proposition <ref>, we see that the natural map

⊗_i=0^r _(M_i, N_i) →_(Ω(M), Ω(N))

is an isomorphism. Since the natural map

⊗_i=0^r _(M_i, N_i) →_^r+1(M, N)

is also an isomorphism (Proposition <ref>), the result follows.




 §.§ The main theorem


The categories ()^ and ()_r are somewhat mysterious; at least, it can be difficult to study them directly. The category (^r+1)^, on ther other hand, is very concrete. The following theorem, and its corollary, can therefore be regarded as a solution to the problem of describing the former two categories.

 
The functor Ω induces an equivalence (^r+1)^→()^.



We first observe that Ω does indeed map (^r+1)^ into ()^. Indeed, since Ω is cocontinuous and the target category is closed under direct limits, it suffices to check on finite length objects. Since Ω is exact and ()^ is closed under extensions, it suffices to check on simple objects. A simple object of (^r+1)^ has the form ^a_0⊠⋯⊠^a_r, and this is sent to ^λ under Ω, where λ is the constraint word of rank r with a(λ)=(a_0, …, a_r). As ^λ belongs to ()^, the statement follows.

We next claim that Ω induces an equivalence ((^r+1)^) →(()^), which will complete the proof by Propsosition <ref>. For a=(a_0,…,a_r) ∈^r+1, let ^a=^a_0⊠⋯⊠^a_r. Using arguments similar to those in <ref>, one can show that the ^a are exactly the indecomposable injectives of (^r+1)^. We have Ω(^a)=^λ, where λ is the constraint word with a(λ)=a. We thus see that Ω induces a well-defined functor

Ω' ((^r+1)^) →(()^).

Since every indecomposable injective of (^r+1)^ is factorizable, Proposition <ref> shows that Ω' is fully faithful. Finally, since every indecomposable injective of ()^ has the form ^λ for λ of rank r, we see that Ω' is essentially surjective. Thus Ω' is an equivalence, and the theorem is proved.


 
We have an equivalence ()_r ≅(^r+1)^.



Combine the previous equivalence with the equivalence ()^≅()_r from Theorem <ref>.


 
Consider the four nearly equivalent categories: (), ()_+, _, and the category of co-semi-simplicial vector spaces. If  is any one of these categories, one can define the level r subquotient _r. For r>0, all four choices yield the same subquotient category. But only for =() is it true that _r is equivalent to _0^⊗ (r+1). Thus, at least from the point of view of this paper, () is the best of these categories. In particular, it is most natural to have the grading for a graded -module to start at 0.




§ KOSZUL DUALITY
 



 §.§ Notation and conventions


We now set some notation and conventions related to homological algebra. We work exclusively with cochain complexes. Thus, if M is a complex then the differential d maps M^n to M^n+1. If M is a complex, we let M[n] be the shifted complex: it has (M[n])^k=M^n+k, and differentials those of M scaled by (-1)^n. We have ^k(M[n])=^n+k(M). If M is a module, regarded as a complex in degree 0, then M[n] is in degree -n.

We say that a bigraded vector space (such as a complex of graded vector spaces or graded -modules, or the homology of such a complex) is degreewise finite if each graded piece is finite dimensional.



 §.§ Tor and minimal resolutions
 

Let M be a graded -module. Define a graded vector space (M) by

(M)_n = M_n/(∑_k=1^n-1 α_k M_n-1 ).

Thus (M)_n records those elements of M_n which cannot be generated by elements of lower degree; one can regard these as minimal generators of M. The functor  is right exact, and its left derived functors exist, as () has enough projectives. We call _i the ith Tor functor.

We say that a projective resolution P_∙→ M is minimal if the differentials in the complex (P_∙) all vanish. It is not difficult to prove that M admits a unique minimal projective resolution, up to isomorphism; the proof is just like that for the analogous fact in commutative algebra, see <cit.>. If P_∙→ M is a minimal resolution then _i (M)=(P_i). Note (^r) is one-dimensional and concentrated in degree r. We thus see that _i (M)_r is the number of ^r summands in the ith term of the minimal projective resolution of M.



 §.§ The Koszul complex
 

We now aim to construct an explicit complex that computes Tor. To motivate our construction, suppose that M is a graded -module. We have a short exact sequence

^n-1 ⊗M_n-1 d_1⟶ M_n ⟶(M)_n →0

where d_1 is given by e_i ⊗ x ↦α_i x. There are some obvious elements of the kernel of d_1, coming from the fundamental relations. To be precise, define

d_2 ^2(^n-1) ⊗M_n-2 →^n-1 ⊗M_n-1,     e_j,i ⊗x ↦e_i ⊗α_j-1 x - e_j ⊗α_i x,

where e_j,i=e_j ∧ e_i for j>i. Then

d_1d_2(e_j,i ⊗x) = α_i α_j-1 x- α_j α_i x = 0.

Thus d_1 ∘ d_2=0.

Continuing in the obvious way, we are lead to the following construction. We define a complex (M) of graded vector spaces. The terms are given by

^-m(M) = ⊕_n ≥m ^m(^n-1) ⊗M_n-m,

where the nth summand has degree n. The differential ^-m(M) →^-m+1(M) is given by

d(e_a_m,…,a_1 ⊗x) = ∑_j=1^m (-1)^j+1 e_a_m,…,â_j,…,a_1 ⊗α_a_j-j+1 x

where 1 ≤ a_1<a_2<⋯<a_m ≤ n-1. Note that 1 ≤ a_j-j+1 ≤ n-m, so α_a_j-j+1 x belongs to M_n-m+1; this verifies that the differential is homogeneous. We call (M) the Koszul complex for the module M. We verify that it is indeed a complex:


Notation as above, we have d^2=0.



We compute:

    d^2(e_a_n,…,a_1⊗ x)
    =    d ( ∑_i=1^n (-1)^i+1 e_a_n,…,â_i,…,a_1⊗α_a_i-i+1 x ) 
    
    =   ∑_i=1^n ∑_j=1^i-1 (-1)^i+j e_a_n,…,â_i,…,â_j,…,a_1⊗α_a_j-j+1α_a_i-i+1 x 
       + ∑_i=1^n ∑_j=i+1^n (-1)^i+j+1 e_a_n,…,â_j,…,â_i,…, a_1⊗α_a_j-j+2α_a_i-i+1 x

The second sum is the negative of the first: to see this, first change the order of the α's (using the fundamental relation), then switch the roles of i and j, and finally change the order of summation. 

We next study the Koszul complex on principal modules. We begin with a lemma.


Let 1 ≤ c_1<⋯<c_m ≤ n be integers, and let a ∈ [n] ∖{c_1,…,c_m}. Then there exists a unique a' ∈ [n-m] such that a=α_c_m⋯α_c_1 a'.



Suppose the result has been proved for m-1, and let us prove it for m. We consider two cases.

First suppose that a<c_m. We have 1 ≤ c_1 < ⋯≤ c_m-1≤ n-1 and a ∈ [n-1] ∖{c_1,…,c_m-1}. By the inductive hypothesis, we can find a' ∈ [(n-1)-(m-1)]=[n-m] such that a=α_c_m-1⋯α_c_1 a'. As a=α_c_ma, we have a=α_c_m⋯α_c_1 a'.

Now suppose that a>c_m. We have 1 ≤ c_1 < ⋯≤ c_m-1≤ n-1 and a-1 ∈ [n-1] ∖{c_1,…,c_m-1}. By the inductive hypothesis, we can find a' ∈ [n-m] such that a-1=α_c_m-1⋯α_c_1 a'. As a=α_c_m(a-1), we have a=α_c_m⋯α_c_1 a', as required.


 
We have ^i((^r))=0 for all r ∈ and i  0.



For r=1, the complex (^1)_n is the complex computing the simplicial homology of the n-2 simplex, and is therefore acyclic away from cohomological degree 0.

We now treat the general case. We claim that the complex (^r)_n is isomorphic to a direct sum of n-1r-1 copies of the complex (^1)_n-r+1, which will prove the proposition. To see this, fix a tuple a=(a_1, …, a_r-1) with 1 ≤ a_1 < ⋯ < a_r-1≤ n-1. Let σ [n-r] → [n-1] be the unique order-preserving injection whose image is disjoint from {a_1,…,a_r-1}. For 1 ≤ b_1<⋯<b_m ≤ n-r, let 1 ≤ a_i' ≤ n-m-1 be the unique integer such that a_i=α_σ(b_m)⋯α_σ(b_1) a_i', which exists by the previous lemma. We define a map of complexes

ϕ_a (^1)_n-r+1 →(^r)_n,

as follows. In cohomological degree -m, it is the map

^m(^n-r) →^m(^n-1) ⊗^r_n-m,     e_b_m,…,b_1 ↦e_σ(b_m),…,σ(b_1) ⊗e_a_1',…,a_r-1',n-m.

(Here we have identified each homogeneous piece of ^1 with .) We now verify that ϕ_a is a map of complexes. We have

d(e_b_m,…,b_1) = ∑_j=1^m (-1)^j+1 e_b_m,…,b̂_j,…,b_1

and

d(e_σ(b_m),…,σ(b_1) ⊗e_a_1',…,a_r-1',n-m) =
∑_j=1^m (-1)^j+1 e_σ(b_m),…,σ(b_j),…,σ(b_1) ⊗α_σ(b_j)-j+1 e_a_1',…,a_r-1',n-m.

It thus suffices to show that

ϕ_a(e_b_m,…,b̂_j,…,b_1) = e_σ(b_m),…,σ(b_j),…,σ_a(b_1) ⊗α_σ(b_j)-j+1 e_a_1',…,a_r-1',n-m.

Let τ=α_σ(b_m)⋯α_σ(b_j)⋯α_σ(b_1) and let a_i” be the unique integer such that a_i=τ a_i”. We have

ϕ_a(e_b_m,…,b̂_j,…,b_1) = e_σ(b_m),…,σ(b_j),…,σ(b_1) ⊗e_a_1”,…,a_r-1”,n-m-1.

It thus suffices to show that a_i”=α_σ(b_j)-j+1 a_j'. We can verify this after appying τ to each side. Thus we must show that a_i=τα_σ(b_j)-j+1 a_i'. But

τα_σ(b_j)-j+1 = α_σ(b_m) ⋯α_σ(b_1),

and so the result follows from the definition of a_i'.

We thus have a map of complexes

ϕ^r-1(^n-1) ⊗(^1)_n-r+1 →(^r)_n,     e_a_1,…,a_r ⊗x ↦ϕ_a(x).

In each cohomological degree, ϕ carries basis elements to basis elements. One easily verifies that it is a bijeciton on basis elements, and so ϕ is an isomorphism of complexes. This completes the proof.


We have defined the Koszul complex (M) of a graded -module M. More generally, suppose that M is complex of graded -modules. We then define

^-m(M) = ⊕_k ∈ ⊕_n ≥m ^k(^n-1) ⊗M^k-m_n-k,

where the (k,n) summand is placed in degree n. The differential is given by

d(e_a_k,…,a_1 ⊗x) = (-1)^n+k e_a_k,…,a_1 ⊗dx + ∑_j=1^k (-1)^j+1 e_a_k,…,a_j+1,a_j-1,…,a_1 ⊗α_a_j-j+1 x,

where x ∈ M^k-m_n-k. This is easily seen to square to 0. We note that we have a natural morphism (M) →(M) in (): this map is zero on ^k(^n-1) ⊗ M^k-m_n-k if k>0, and the quotient map M^-m_n →(M^-m_n) if k=0.

 
The functor  induces a functor (()) →(), which is the left derived functor of .



The functor  satisfies the hypotheses of Proposition <ref>, and so, by that proposition, we see that  takes quasi-isomorphisms to quasi-isomorphisms. It thus induces a functor on the derived categories. Suppose now that M is a bounded above complex in (), and choose a quasi-isomorphism P → M where P is a bounded above complex of projectives. Then the left derived functor of  is defined at M and equal to (P). Since the maps (M) ←(P) →(P) are quasi-isomorphisms (for the second map, this follows from Proposition <ref>), we see that (M) is the value of the derived functor as well. This shows that  is the left derived functor of  on the bounded above category. The general case can be deduced from a limiting argument (every complex is a filtered colimit of bounded above complexes, and both  and  commute with filtered colimits); we will not actually need this, so we omit the details.


 
Let M be a graded -module. Then _i (M) ≅^-i((M)).




 §.§ The duality functor
 

For 1 ≤ k ≤ n, let T_k ^n+1→^n be the map given by

T_k(e_i_1,…,i_n+1) = e_i_1,…,i_k-1,i_k+1,…,i_n+1.

The T_k give a basis for _(^n+1, ^n). Given a complex M ∈(()) we now define another complex (M) ∈(()). The terms are given as follows:

^-m(M) = ⊕_k ∈ (M_k^m-k)^* ⊗^k.

Here, (M_k^m-k)^* is simply regarded as an ungraded vector space; the () structure on ^-m(M) comes entirely from ^k. For λ∈ (M_k^m-k)^* and x ∈^k, the differential is given by

d(λ⊗x) =(-1)^k d^*(λ) ⊗x + ∑_i=1^k-1 (-1)^i+1 α_i^*(λ) ⊗T_i(x)

where d^* is “naive” dual to the differential on M (i.e., we do not introduce any signs). We note that the differential is a morphism in the category (). We verify below (Lemma <ref>) that this is actually a complex. Granted this, we have thus defined a functor

(())^ →(()).

If this construction seems capricious, see <ref> for an explanation of its origin. We note that when M is a graded -module, regarded as a complex in degree 0, the complex (M) simplifies to 

⋯→M_2^* ⊗^2 →M_1^* ⊗^1 →M_0^* ⊗^0 →0 →⋯
with differential

d(λ⊗x) = ∑_i=1^k-1 (-1)^i+1 α_i^*(λ) ⊗T_i(x),     λ⊗x ∈M_k^* ⊗^k.

The main result of this section is the following theorem:

 
The functor  induces a contravariant functor (()) →(()). Moreover, there is a canonical natural transformation M →((M)) in (()) that is an isomorphism if ^∙(M) is degreewise finite.


The following lemma shows that (M) is just the Kozul complex (M) with the terms shuffled around and dualized.

 
Let M ∈(()).


  * We have a canonical isomorphism of vector spaces ^-m(M)_n ≅ (^m-n(M)_n)^*.

  * The isomorphism from (a) is compatible with the differentials, in the sense that the diagram


^-m(M)_n [r]^d @=[d]     ^-m+1(M)_n @=[d] 

(^m-n(M)_n)^* [r]^d^*     (^m-n-1(M)_n)^* 

commutes.

  * (M) is a complex, that is, d^2=0.

  * We have an isomorphism (M)_n ≅(M)_n^*[n] in ().

  * We have an isomorphism of vector spaces ^-m((M))_n ≅^m-n((M))_n^*.




(a) We have

^-m(M)_n = ⊕_k ∈ (M_k^m-k)^* ⊗^k_n

and

^m-n(M)_n = ⊕_k ∈ ^n-k(^n-1) ⊗M^m-k_k.

In the second sum, we have changed k to n-k. For notational ease, put

D^m,n,k=(M_k^m-k)^* ⊗^k_n,     K^m,n,k=^n-k(^n-1) ⊗M^m-k_k

so that ^-m(M)_n is the sum of the D^m,n,k and ^m-n(M)_n is the sum of the K^m,n,k. We note that D^m,n,k=K^m,n,k=0 if k<0 or k>n. In particular, the above direct sums are finite.

Now, ^k_n has a basis consisting of elements e_i_1,…,i_k with 1 ≤ i_1 < ⋯ < i_k=n. We identify this with ^k-1(^n-1) in the obvious manner (discard i_k). Finally, ^k-1(^n-1) and ^n-k(^n-1) admit a canonical perfect pairing to the one-dimensional space ^n-1(^n-1). Via the isomorphism μ^n-1(^n-1) → mapping e_1,2,…,n-1 to 1, this identifies ^k-1(^n-1) with the dual of ^n-k(^n-1) and vice versa. We thus see that the terms in the sum for ^-m(M)_n are identified with the terms in the sum for ^m-n(M)_n. Since both sums are finite, this identifies ^-m(M)_n with the dual of ^m-n(M)_n.

(b) Let

⟨, ⟩^-m(M)_n ×^m-n(M)_n →.

be the perfect pairing from part (a). Explicitly, given λ⊗ v ∈ D^m,n,k, with λ∈ (M_k^m-k)^* and v ∈^k_n ≅^k-1(^n-1), and w ⊗ x ∈ K^m,n,ℓ, with w ∈^n-ℓ(^n-1) and x ∈ M_ℓ^m-ℓ, we have

⟨λ⊗v, w ⊗x ⟩= δ_k,ℓ μ(v ∧w) (λ, x),

where (,) is the canonical pairing between M^m-k_k and its dual.

To prove that the isomorphism of  with the dual of  is compatible with the differentials, it is equivalent to show that the differentials are adjoint with respect to the pairing. Thus let λ⊗ v ∈ D^m,n,k and w ⊗ x ∈ K^m-1,n,ℓ be given. We must show

⟨d(λ⊗v), w ⊗x ⟩= ⟨λ⊗v, d(x ⊗w) ⟩.

It suffices to treat the case where v and w are pure tensors, say v=e_a_1,…,a_k-1 with 1 ≤ a_1 < ⋯ < a_k-1≤ n-1 and w=e_b_1,…,b_n-ℓ with 1 ≤ b_1 < ⋯ < b_n-ℓ≤ n-1. We have

d(λ⊗v) = (-1)^k d^*(λ) ⊗v + ∑_i=1^k (-1)^i+1 α_i^*(λ) ⊗e_a_1,…,â_i,…,a_k-1.

Now, the first term above belongs to D^m-1,n,k and the second belongs to D^m-1,n,k-1. We thus have

⟨d(λ⊗v), w ⊗x ⟩= 
(-1)^k μ(v ∧w) ( λ, dx)     if k=ℓ 

∑_j=1^k (-1)^j+1 μ(e_a_1,…,â_j,…,a_k-1,b_1,…,b_n-ℓ) (λ, α_j x)     if k=ℓ+1 

0     otherwise


On the other hand, we have

d(w ⊗x) = (-1)^ℓ w ⊗dx + ∑_i=1^n-ℓ (-1)^i+1 e_b_1,…,b̂_i,…,b_n-ℓ ⊗α_b_i-i+1 x.

The first term belongs to K^m,n,ℓ and the second belongs to K^m,n,ℓ+1. We thus see that

⟨λ⊗v, d(w ⊗x) ⟩= 
(-1)^k μ(v ∧w) (λ, dx)     if k=ℓ 

∑_i=1^n-ℓ (-1)^i+1 μ(e_a_1,…,a_k,b_1,…,b̂_i,…,b_n-l) (λ, α_b_i-i+1 x)     if k=ℓ+1 

0     otherwise


If k ℓ+1 then we clearly have our desired equality. Thus let us now assume k=ℓ+1. We must prove

∑_j=1^ℓ+1 (-1)^j+1 μ(e_a_1,…,â_j,…,a_ℓ,b_1,…,b_n-ℓ) (λ, α_j x)
= ∑_i=1^n-ℓ (-1)^i+1 μ(e_a_1,…,a_ℓ+1,b_1,…,b̂_i,…,b_n-ℓ) (λ, α_b_i-i+1x).

The only way for either side to be non-empty is if the sets {a_1,…,a_ℓ+1} and {b_1,…,b_n-ℓ} have exactly one element in common. Thus suppose this is the case, and that a_j=b_i. Then it is the jth term on the left and ith term on the right that are possibly non-zero, so we must show that they coincide. Now, we have

{1,…,b_i} = {1,…,a_j} ⨿{1,…,b_i-1},

and so, counting, we find b_i=j+i-1. Thus α_b_i-i+1=α_j. Furthermore,

μ(e_b_n-ℓ,…,b̂_i,…,b_1,a_1,…,a_ℓ+1)=(-1)^i+j μ(e_b_n-ℓ,…,b_1,a_1,…,â_j,…,a_ℓ).

as it takes i+j-2 transpositions to transform the subscript sequence on left to the one on the right: it clearly takes zero transpositions if i=j=1, and one more every time i or j is incremented. This proves our desired equality.

(c–e) These statements follow easily from parts (a) and (b).



The lemma implies that ⊕_n ≥ 0^m-n((M))^*_n canonically carries the structure of a graded -module, as it is naturally isomorphic to ^-m((M)). When M is a module, this is the dual of the mth linear strand of the minimal resolution of M.



The functor  takes quasi-isomorphisms to quasi-isomorphisms. In particular, it induces a functor on the derived category.



Let f  M → N be a quasi-isomorphism in (()). For any m,n ∈, the diagram


^-m((N))_n [r] @=[d]     ^-m((M))_n @=[d] 

^m-n((N))_n^* [r]     ^m-n((M))_n^* 

commutes, where the vertical isomorphisms are those from Lemma <ref> and the horizontal maps are induced by f. Since  takes quasi-isomorphisms to quasi-isomorphisms, it follows that the bottom arrow is an isomorphism. Thus the top arrow is as well, which implies that (f) (N) →(M) is a quasi-isomorphism.


 
We have (^n) ≅^n[n] and (^n)=^n[n].



It follows directly from the definition that ^-m(^n) is zero unless m=n, and it is then ^n. Thus ^-m(^n)=^n[n], as claimed.

From Lemma <ref>, we have ^-m((^r))_n ≅^m-n((^r))_n^*. By Proposition <ref>, ^i((^r))_n is one-dimensional if i=0 and n=r, and vanishes otherwise. Thus ^-m((^r))_n is one-dimensional if m=n=r, and vanishes otherwise. It follows that (^r)=^r[r], as claimed.


For a complex of graded -modules M, define a complex (M) as follows. The terms are given by

    (M)^n = ⊕_k,ℓ∈ M^k+n-ℓ_ℓ⊗ (^ℓ_k)^* ⊗^k.

For x ∈ M_ℓ^k+n-ℓ and λ∈ (^ℓ_k)^* and y ∈^k, the differential is given by

    d(x ⊗λ⊗ y) =   
    (-1)^k+ℓ dx ⊗λ⊗ y +
    (-1)^k ∑_i=1^ℓ[ (-1)^i+1α_i(x) ⊗ T_i^*(λ) ⊗ y ] 
       + ∑_j=1^k-1[ (-1)^j+1 x ⊗α_j^*(λ) ⊗ T_j(y) ].

We note that (M) is exactly the complex one obtains by computing ((M)) and formally moving duals over tensor products and removing double duals.


 
For M ∈(()) we have a canonical injection (M) →((M)). If M is degreewise finite then this map is an isomorphism of complexes.



This follows from the description of (M) given above.


 
For M ∈(()) we have a canonical quasi-isomorphism (M) → M.



Let ϵ (^k_k)^* → be the isomorphism taking the standard basis vector of (^k_k)^* to 1 ∈. For a graded -module N, there is a canonical map ^k ⊗ N_k → N of graded -modules stemming from the mapping property of ^k. We let y ∗ x ∈ N denote the image of y ⊗ x ∈^k ⊗ N_k.

Define a map f (M) → M as follows. Let x ⊗λ⊗ y be an element of (M)^n, using notation as in (<ref>). We define:

f(x ⊗λ⊗y) = 
0     if k ℓ 

(-1)^k2 ·ϵ(λ) ·(y ∗x)     if k=ℓ 

We claim that f is a map in the category (()). On each term, it is a map in () by definition, so it suffices to prove that f commutes with the differentials. Thus let x ⊗λ⊗ y as above be given. We proceed in four cases:

Case 1: k<ℓ. There is nothing to prove, as ^ℓ_k=0.

Case 2: k=ℓ. In (<ref>), we have T_i^*(λ) ∈ (^k+1_k)^*=0 and α_j^*(λ) ∈ (^k_k-1)^*=0; furthermore, (-1)^k+ℓ=1 since k=ℓ. We thus find

d(x ⊗λ⊗y) = dx ⊗λ⊗y.

Hence,

df(x ⊗λ⊗y) = (-1)^k2 ·ϵ(λ) ·d(y ∗x),     f(d(x ⊗λ⊗y))=(-1)^k2 ·ϵ(λ) ·(y ∗dx).

These quantites are equal, since the map ∗^k ⊗ N_k → N is functorial in the graded -module N.

Case 3: k=ℓ+1. We have f(x ⊗λ⊗ y)=0, and so we must show that f(d(x ⊗λ⊗ y))=0. Now, in (<ref>), the first time is killed by f, since λ∈ (^k-1_k)^*. We thus have

    f(d(x ⊗λ⊗ y))
    =    (-1)^k+k2∑_i=1^k-1[ (-1)^i+1ϵ(T_i^*(λ)) · (y ∗α_i(x)) ] 
       + (-1)^k-12∑_j=1^k-1[ (-1)^j+1ϵ(α_j^*(λ)) · (T_j(y) ∗ x) ]

The signs are always opposite of each other. It thus suffices to show

ϵ(T_i^*(λ))=ϵ(α_i^*(λ)),     y ∗α_i(x) = T_i(y) ∗x,

which we leave to the reader.

Case 4: k>ℓ+1. Both x ⊗λ⊗ y and d(x ⊗λ⊗ y) are killed by f, as follows directly from the definition.


We now show that f is a quasi-isomorphism. By Proposition <ref>, it is enough to treat the case where M=^r, regarded as a complex concentrated in degree 0. We thus assume this is the case. By Lemma <ref> and Proposition <ref>, we have (^r) ≅((^r)) ≅^r in (()). Thus to verify the claim, it suffices to show that the map ^0(f) ^0((^r)) →^r is an isomorphism. Since ^0((^r)) ≅^r, it is enough to show that this map is surjective in degree r. Since (^r)^1=0, it suffices to show that f (^r)^0_r →^r_r is surjective. This is clear from the definition of f.


 
Let W ∈() be a chain complex of vector spaces such that each cohomology group is finite dimensional, and let U ⊂ W be a subcomplex such that each term is finite dimensional. Then there exists an intermediate complex U ⊂ V ⊂ W such that each term of V is finite dimensional and the inclusion V → W is a quasi-isomorphism.



Left to the reader.


 
Let M ∈(()). Suppose that ^∙(M) is degreewise finite. Then there exists a subcomplex N ⊂ M that is degreewise finite and for which the inclusion is a quasi-isomorphism.



We construct a chain N_0 ⊂ N_1 ⊂⋯⊂ M of subcomplexes with the following properties: (a) each term of N_n is finitely generated (and thus degreewise finite); (b) N_n → M is quasi-isomorphism in degrees ≤ n; (c) N_n and N_n+1 are equal in degrees ≤ n. Taking N=⋃_n ≥ 0 N_n gives the requisite subcomplex.

Suppose N_n-1 has been constructed; for n=0 we take N_n-1=0. We now construct N_n. Appealing to Lemma <ref>, choose a subcomplex V of M_n containing (N_n-1)_n such that the terms of V are finite dimensional and the inclusion V → M_n is a quasi-isomorphism. We then take N_n to be the sum (inside of M) of N_n-1 and the complex of submodules generated by V. Clearly, N_n coincides with N_n-1 in degrees ≤ n-1, and N_n coincides with V in degree n. Thus it has the required properties.



Let M ∈(()). By Lemmas <ref> and <ref>, we have maps

M ←(M) →((M)).

Since the first map is a quasi-isomorphism, this diagram defines a morphism M →((M)) in the derived category. To complete the proof, it suffices to show that the second map is a quasi-isomorphism when ^∙(M) is degreewise finite. Thus suppose this is the case. By Lemma <ref>, we can find a subcomplex N ⊂ M that is degreewise finite and for which the inclusion is a quasi-isomorphism. Now consider the following diagram:


(N) [r] [d]     ((N)) [d] 

(M) [r]     ((M)) 

Since  and  take quasi-isomorphisms to quasi-isomorphisms, the vertical maps are quasi-isomorphisms. By Lemma <ref>, the top map is a quasi-isomorphism. Thus the bottom map is as well.




 §.§ Duality and the concatenation product


We now investiage how the duality functor interacts with the concatenation product. Given complexes M and N of graded -modules, we let M ⊙ N be the complex with

(M ⊙N)^n = ⊕_i+j=n M^i ⊙N^j

and differential

d(x ⊙y) = dx ⊙y + (-1)^i x ⊙dy

for x ∈ M^i and y ∈ N^j.

 
For any M,N ∈(()) we have a canonical morphism

(M) ⊙(N) →(M ⊙N)

in (()). It is an isomorphism if at least one of M or N is degreewise finite.



For K ∈(()), we have

(K) = ⊕_n,r (K^n_r)^*[n+r] ⊗^r,

as a graded object of (). In this form, the differential is given by

d(λ⊗x) = (-1)^r d^*(λ) ⊗x + ∑_i=1^r-1 (-1)^i+1 α_i^*(λ) ⊗T_i(x)

for λ∈ (K^n_r)^* and x ∈^i. We will use this description of  in this proof.

We have

(M) ⊙(N) = ⊕_m,n,r,s ((M^m_r)^*[m+r] ⊗^r) ⊙((N^n_s)^*[n+s] ⊗^s).

On the other hand,

(M ⊙N) = ⊕_m,n,r,s (M^m_r ⊗N^n_s)^*[m+n+r+s] ⊗^r+s.

We thus have a map

ϕ(M) ⊙(N) →(M ⊙N)

given by

ϕ(λ⊗x ⊙μ⊗y) = ϵ^m,n_r,s (λ⊙μ) ⊗(x ⊙y),

for λ∈ (M^m_r)^*, μ∈ (N^n_s)^*, x ∈^r, and y ∈^s. Here λ⊙μ denotes the natural element of (M^m ⊙ N^n)^* and x ⊙ y ∈^r+s=^r ⊙^s, and ϵ^m,n_r,s denotes a sign to be determined. It is clear that, on each term, ϕ is a map of graded -modules, and that it is an isomorphism if at least one of M or N is degreewise finite. To complete the proof, it suffices to show that ϕ is compatible with differentials.

Let λ, μ, x, and y be as above. In what follows, we omit the symbols ⊗ and ⊙. Regarding λ x μ y ∈(M) ⊙(N), we have

    d(λ x μ y)
    =    d(λ x) μ y + (-1)^m+rλ x d(μ y) 
    
    =    (-1)^r d^*(λ)xμ y+∑_i=1^r-1 (-1)^i+1α_i^*(λ) T_i(x) μ y 
       + (-1)^m+r+sλ x d^*(μ) y + ∑_j=1^s-1 (-1)^m+r+j+1λ x α_j^*(μ) T_j(y)

and so

    ϕ(d(λ x μ y))
    =    (-1)^r ϵ^m-1,n_r,s d^*(λ)μ xy+∑_i=1^r-1 (-1)^i+1ϵ^m,n_r-1,sα_i^*(λ) μ T_i(x) y 
       + (-1)^m+r+sϵ^m,n-1_r,sλ d^*(μ) xy + ∑_j=1^s-1 (-1)^m+r+j+1ϵ^m,n_r,s-1λα_j^*(μ) x T_j(y).

On the other hand, regarding λμ x y ∈(M ⊙ N), we have

d(λμx y) = (-1)^r+s d^*(λμ) xy + ∑_k=1^r+s-1 (-1)^k+1 α_k^*(λμ) T_k(xy).

We now come to the key calculations:

α_k^*(λμ) = 
α_k^*(λ) μ    if 1 ≤ k ≤ r-1 

λα_k-r^*(μ)     if r+1 ≤ k ≤ r+s-1 

0     if k=r 

and

T_k(xy) = 
T_k(x) y     if 1 ≤ k ≤ r-1 

x T_k-r(y)     if r+1 ≤ k ≤ r+s-1 

?     if k=r. 

The question mark indicates that there is not a nice formula. The proofs are easy and left to the reader. We thus find

    ϵ^m,n_r,s d(ϕ(λ x μ y))
    =    (-1)^r+s d^*(λ) μ xy + (-1)^r+s+mλ d^*(μ) xy 
       + ∑_i=1^r-1 (-1)^i+1α_i^*(λ) μ T_i(x) y + ∑_j=1^s-1 (-1)^r+j+1λα_j^*(μ) x T_j(y).

We now see that by choosing ϵ^m,n_r,s = (-1)^ms we obtain ϕ d = d ϕ.


For a constraint word λ, define the conjugate word c(λ) to be the constraint word with b changed to a and vice versa. Combining the proposition with Proposition <ref>, we obtain:

 
We have (^λ) ≅^c(λ)[ℓ], where ℓ is the length of λ.




 §.§ The finiteness theorem


Theorem <ref> is essentially a formality: its proof does not rely on any difficult results about -modules. By contrast, the following theorem relies on one of our main theorems:

 
The duality functor  carries ^b_(()) into itself.



By Theorem <ref>, ^b_(()) is generated (as a triangulated category) by the standard objects ^λ. It therefore suffices to show that (^λ) belongs to ^b_(()) for all λ. This follows from Corollary <ref>.


 
The duality functor induces an equivalence

^b_(())^ →^b_(())

whose square is canonically isomorphic to the identity functor.



By Corollary <ref>,  induces a self-map of (), which we still denote by . From Corollary <ref>, we see that [^λ] = (-1)^ℓ(λ) [^c(λ)]. Identifying () with {a,b}, we see that  is the ring automorphism taking a to -b and b to -a.




 §.§ Betti tables
 

Let M be a graded -module. By analogy with commutative algebra, we define the Betti table of M, denoted β(M), to be the two-dimensional array given by

β(M)_i,j = _j (M)_i+j.

Thus the entry in row i and column j records the number of generators in degree i+j of the jth term in the minimal resolution of M. We define the (Castelnuovo–Mumford) regularity of M, denoted ρ(M), to be the minimum n such that β_i,j(M)=0 for all i>n and all j ∈, or ∞ if no such n exists; thus ρ(M) is the index of the final non-zero row in β(M). The following theorem is our main result on these invariants:

 
Let M be a finitely generated graded -module. Then ρ(M) is finite, that is, β(M) has only finitely many non-zero rows. Furthermore, each row is eventually polynomial, that is, for each i there is a polynomial p_i(t) ∈[t] such that β(M)_i,j=p_i(j) for j ≫ 0.



By Corollary <ref> and Lemma <ref>(e) we have

β(M)_i,j= _j (M)_i+j =^-j((M))_i+j =^-i((M))_i+j

By Theorem <ref>, ^-i((M)) is a finitely generated graded -module for all i, and non-zero for only finitely many i. We thus see that β(M) has only finitely many non-zero rows. Furthermore, for fixed i, we see that β(M)_i,j is the dimension of the i+j graded piece of the finitely generated graded -module ^-i((M)), and thus is eventually a polynomial in j by Theorem <ref>.
















































 §.§ Conceptual explanation of  and 
 

We now give a conceptual explanation for some of the constructions appearing in this section. Let A denote the shuffle algebra _(⟨ 1 ⟩), as defined in <ref>. Recall that _A is essentially equivalent to the category of graded -modules; for the present discussion, we ignore the minor difference between the two categories. If M is an A-module then the usual construction yields a Koszul complex ^∙_(⟨ 1 ⟩) ⊗ M. Translating this back across the equivalence of categories yields our Koszul complex (M). Furthermore, the usual form of Koszul duality yields a derived equivalence between A and B, where B=_(⟨ 1 ⟩) is the exterior algebra on ⟨ 1 ⟩. It so happens that A and B, while not isomorphic as algebras, are Morita equivalent, that is, their module categories are equivalent; one constructs an equivalence simply by playing with signs. This yields a second equivalence between the derived categories of A and B. Combining the two produces a derived auto-equivalence of A. Translating this back to () yields our duality functor .

Thus, from the perspective of shuffle algebras, the rather opaque constructions we made above become somewhat more clear. We chose to stick with the language of -modules, however, to maintain consistency with the rest of the paper.



§ GROTHENDIECK GROUPS REVISITED
 



 §.§ The pairing on ()
 

Let  be a -linear abelian category with Grothendieck group (). Suppose that the following condition holds:


  (∗) For all objects M,N ∈ the -vector space ^i_(M,N) is finite dimensional and vanishes for i ≫ 0.

Then () admits a canonical pairing, as follows:

⟨[M], [N] ⟩= ∑_i ≥0 (-1)^i ^i_(M, N).

We have seen (Propositions <ref> and <ref> and Corollary <ref>) that both ()^ and ()^ satisfy condition (∗), and so () and () admit pairings.  We let ⟨ , ⟩ denote the pairing on () and (,) the pairing on ().



 §.§ Higher multiplicities
 

Let λ be a constraint word of rank r. Then T_≥ r(^λ) is a simple object of the category ()_≥ r (Proposition <ref>), and so there is an associated multiplicity function _λ (see <ref>). We define _λ on () by simply composing with T_≥ r, that is, for M ∈() we put _λ(M)=_λ(T_≥ r(M)). It follows directly from the definition that _λ(M)=0 if (M)<r. We similarly define μ_λ in the smooth case.

 
Let λ be a constraint word and let M be a graded -module. Then

_λ(M)=_(M, ^λ) = ⟨[M], [^λ] ⟩
In particular, if M is finitely generated then _λ(M) is finite. The analogous statements holds in the smooth case.



It follows from the theory in <ref> that T_≥ r(^λ) is the injective envelope of T_≥ r(^λ), where r is the rank of λ. We thus have:

    _λ(M)
       = (T_≥ r(M), T_≥ r(^λ)) 
       = _(M, _≥ r(^λ)) 
       = _(M, ^λ) 
       = ⟨ [M], [^λ] ⟩.

The first step follows from Proposition <ref>; the second uses the adjunction between T_≥ r and S_≥ r: the third uses the fact that ^λ is saturated (which follows from the theory in <ref>); and the fourth step simply uses the fact that ^λ is injective, so that all higher 's into it vanish.



The multiplicity function _λ induces a homomorphism _λ() →.



If λ=^n then _λ is the multiplicity function _n already studied.




 §.§ The non-commutative Hilbert series
 

Let M be a finitely generated graded -module. We define the non-commutative Hilbert series of M by

_M = ∑_λ _λ(M) λ,

where the sum is over all constraint words λ. This is a formal -linear combination of words, and thus an element of the ring a,b of non-commutative power series. We note that _M becomes the usual Hilbert series _M(t) under the substitutions a → 0 and b → t. We also note that if M has homogeneous level ≤ r then _λ(M)=0 if λ has rank >r, and so every monomial appearing in _M(s,t) with non-zero coefficient has a appearing at most r times. The invariant  obviously factors through the Grothendieck group; thus for x ∈(), or x ∈{a,b}, we have an associated quantity _x.

For a finitely generated smooth -module M, we define

_M = ∑_λ μ_λ(M) λ.

Similar comments to the above apply to this construction.

 
Let M be a finitely generated smooth -module. Then _M=_Ξ(M).



Using the adjunction (Ξ, Φ) (Proposition <ref>), we have

_λ(Ξ(M)) = _(Ξ(M), ^λ) = _(M, ^λ) = μ_λ(M),

and so the result follows.


Every element f of a,b can be written uniquely in the form f=x+ay+bz with x ∈ and y,z ∈ a,b. Let

_M = _0(M) + a ^a_M + b ^b_M

be this decomposition of _M. Recall (<ref>) that we have constructed several operators on the Grothendieck groups, including (-)^†, ξ, ψ, and σ.

 
Let κ be the endomorphism of () given by κ(x)=(ξ(ψ(x^†)))^†. Then for any x ∈(), we have

_x^ = _κ(x)

and

_x^ = _σ(x) - _0(x) - a ^a_x.




By Proposition <ref>, we have

^a λ=^1 ⊙^λ=Γ(Φ((^λ)^†))^†.

Thus, for a graded -module M we have

_(M, ^aλ)=_(Ξ(Ψ(M^†))^†, ^λ),

where we have used the adjunctions (Ψ, Γ) and (Ξ, Φ) (Propositions <ref> and <ref>). It follows that _aλ(x)=_λ(κ(x)) for x ∈(), and so

^a_x = ∑_λ _a λ(x) λ= ∑_λ _λ(κ(x)) λ= _κ(x),

which proves the first identity.

We now prove the second. Let λ be a constraint word. It follows from the proof of Lemma <ref> that

(^λ) = 
^bλ     if λ has first letter b 

^bλ ⊕^λ     otherwise. 

Thus, using the adjunction (Σ, ) (Proposition <ref>), we see that for a graded -module M we have

_(Σ(M), ^λ) = 
_(M, ^bλ)     if λ has first letter b 

_(M, ^bλ ⊕^λ)     otherwise. 

Thus for x ∈(), we have

_bλ(x) = _λ(σ(x)) - 
0     if λ has first letter b 

_λ(x)     otherwise. 

We thus have

^b_x = ∑_λ _bλ(x) λ=∑_λ _λ(σ(x)) - _0(x) - ∑_λ=a ν _λ(x) λ
As the first term is _σ(x) and the third is a^a_x, the result follows.


 
Let x ∈{a,b}. Then

_bx=b _x.

Furthermore, assuming x=1 or x ∈ b {a,b} and n ≥ 1, we have

_a^n x = a _(1+⋯+a^n-1)x+b (1-b)^-1 _a^n-1x.




We have κ(bx)=(ξ(ψ(x^† b)))^†=0 by Proposition <ref>, and so ^a_bx=0 by Proposition <ref>. As σ(bx)=x (Proposition <ref>), we find ^b_bx=_x by Proposition <ref>. Finally, we have _0(bx)=0 (obvious), and so we conclude _bx=b ^b_bx=b _x.

Now suppose x=1 or x ∈ b{a,b} and n ≥ 1. Then

κ(a^n x)=(ξ(ψ(x^† a^n))^†=(ξ(x^† a^n-1))^†=(1+⋯+a^n-1) x

(Proposition <ref>). Thus ^a_a^nx=_(1+⋯+a^n-1) x by Proposition <ref>. We have σ(a^n x)=a^n x+a^n-1 x (Proposition <ref>), and so

^b_a^n x=_a^n x+_a^n-1 x-_0(a^nx)-a _a^n x^a=b^b_a^n x+_a^n-1 x.

Thus ^b_a^n x=(1-b)^-1_a^n-1 x. As _0(a^nx)=0, we have

_a^nx=a^a_a^nx+b^b_a^nx=a _(1+⋯+a^n-1)x+b(1-b)^-1 _a^n-1x,

which proves the proposition.


Proposition <ref> gives recursive formulas that allow one to calculation _x for any x ∈{a,b} in finitely many steps. These formulas also yield the following theorem:

 
Let M be a finitely generated graded -module. Then _M is a non-commutative rational function. More precisely, it is a finite sum of terms of the form f_0(b) a f_1(b) a ⋯ a f_k(b) where f_0, …, f_k are rational functions of b with denominators a power of 1-b. The analogous statements hold in the smooth case.



The statement in the graded case follows immediately from Proposition <ref>, while the smooth case follows from the graded case and Proposition <ref>.



Using Proposition <ref>, we find

_aba=aba+ab^2(1-b)^-1+b^2(1-b)^-1a+b^3(1-b)^-2.

The coefficient of ab^n is 0 if n=0,1 and 1 if n ≥ 2. We thus see that ^a and ^ab have multiplicity 0 in ^aba, while ^ab^n has multiplicity 1 in ^aba for n ≥ 2. (Technically, these multiplicities should be taken in the quotient category ()_≥ 1.)




 §.§ Another non-commutative series
 

For x ∈{a,b}, define

_x = ∑_λ ⟨x, λ⟩λ,     _x = ∑_λ (x, λ) λ.

Arguing as in the previous section, one can show

_a^nx=a_(1+⋯+a^n-1)x+b_a^n-1x,     _bx = b(1+b)^-1 _x,     _x=_ξ(x),

where in the first formula we assume x=1 or x ∈ b{a,b} and n ≥ 1. These formulas allow one to compute _x and F_x in finitely many steps, and thus the pairings ⟨ x, λ⟩ and (x, λ) as well.


We have

_aba=(a+b) b(1+b)^-1 (a+b).

The coefficient of ab^n here is 0 if n=0,1 and (-1)^n if n ≥ 2. It follows that

∑_i ≥0 (-1)^i ^i_(^aba, ^ab^n) = 
0     if n=0,1 

(-1)^n     if n ≥ 2 




One can also consider the following variant series:

'_x = ∑_λ ⟨λ, x ⟩λ,     '_x = ∑_λ (λ, x ) λ
The duality functor  induces an involution δ on () that satisfies ⟨δ(x), δ(y) ⟩ = ⟨ y, x ⟩. Thus '_x is be obtained from _δ(x) under the substitutions a → -b and b → -a. The adjunction (Φ, F) discussed in <ref> gives '_x='_y where y=σ(γ(x)^†)^†. Thus one can effectively compute '_x and '_x.




 §.§ Effectivity
 

Let  be an abelian category. Every element of the Grothendieck group () can be written in the form [M]-[N] where M and N are objects of . We say that an element () is effective if it has the form [M] for some object M of . The collection of effective elements forms a submonoid of (). Characterizing this submonoid can be a difficult problem in general. We now solve this problem for () and ().

 
An element x of () is effective if and only if _λ(x) ≥ 0 for all constraint words λ. The analogous result holds in the smooth case.



In this proof, we say that x ∈() is -positive if _λ(x) ≥ 0 for all constraint words λ. For a finitely generated graded -module M, the quantity _λ(M) is non-negative, and so any effective element of () is -positive. We now prove the converse. For r ∈, consider the following statement:


  S(r): Let x ∈() be effective and let y ∈()_≤ r be such that x-y is -positive. Then x-y is effective.

Clearly, it suffices to prove S(r) for all r. We proceed by induction on r; note that S(r) is trivially true for r<0.

Let r ≥ 0 be given and suppose S(r-1) holds. To prove S(r), it suffices to treat the case where y=[^λ] for a constraint word λ of rank r, as ()_≤ r is spanned by these classes and ()_≤ r-1. Thus suppose that M is a finitely generated graded -module such that [M]-[^λ] is -positive. We will show that it is effective.

We have

_λ([M]-[^λ])=_λ(M)-1 ≥0,

and so _λ(M) ≥ 1. We thus see (Proposition <ref>) that there is a non-zero map f  M →^λ. Since T_≥ r(^λ) is an essential extension of T_≥ r(^λ), it follows that K=f(M) ∩^λ is non-zero. Now, we have

[M]=[(f)]+[(f)/K]+[K],

and

[^λ]=[K]+[^λ/K].

We thus see that

[M]-[^λ] = [(f) ⊕(f)/K] - [^λ/K].

This element is -positive by assumption. Since the first term on the right is effective and the second belongs to ()_≤ r-1 (by Theorem <ref>), this element is thus effective by S(r-1). This completes the proof in the graded case, and the same argument applies in the smooth case.



The effective submonoid of () is saturated: that is, if x ∈() and nx is effective for some positive integer n then x is effective. Similarly in the smooth case.



Suppose nx is effective. Then _λ(nx)=n _λ(x) ≥ 0, and so _λ(x) ≥ 0. Thus x is effective.



Theorem <ref> on its own is not quite an effective test for effectivity, as it requires verifying infinitely many conditions. However, combining Theorem <ref> with Proposition <ref> gives an effective test, as follows. Let x ∈() ≅{a,b}, and suppose we are given x in the form ∑_i=1^n n_i λ_i where the n_i are integers and the λ_i are constraint words. Theorem <ref> states that x is effective if and only if every coefficient of _x is non-negative. Using Proposition <ref>, we can compute _x in finitely many steps. In fact, it yields an expression for _x as a finite sum of terms of the form f_0(b)a⋯ a f_k(b), where each f_k(b) is given as a polynomial of b divided by some power of 1-b. We can then test, in finitely many steps, if all coefficients in the series expansion of this expression are non-negative.



The isomorphism ϕ() →() carries effective elements to effective elements, however, it is not a bijection between the sets of effective elements. Indeed, [^1]-[^0] is an effective element of () as it is the class of the kernel of the augmentation map ϵ^1 →^0. However, [^1]-[^0] is not an effective element of (): indeed, it has _0=-1.




§ CATEGORICAL BACKGROUND
 



 §.§ Multiplicities
 

Let  be an abelian category and let L be a simple object of . Let _n be the class of all objects M of  for which there exists a chain

F_1 ⊂G_1 ⊂⋯⊂F_n ⊂G_n ⊂M

with G_i/F_i ≅ L for all i. Clearly, we have _n+1⊂_n. We define the multiplicity of L in M, denoted μ_L(M), to be the maximum n for which M ∈_n, or ∞ if M belongs to _n for all n.


We have the following:


  * μ_L is additive in short exact sequences, that is, given a short exact sequence

0 →M_1 →M_2 →M_3 →0

we have μ_L(M_2)=μ_L(M_1)+μ_L(M_3).

  * If N is a subquotient of M then μ_L(N) ≤μ_L(M).

  * Suppose  satisfies (AB5). If M=⋃_i ∈ I M_i (directed union) then we have μ_L(M)=sup_i ∈ Iμ_L(M_i).




(a) It is clear that μ_L(M_3) ≥μ_L(M_1)+μ_L(M_3): indeed, if M_1 ∈_a and M_3 ∈_b then we can splice together the given chains in M_1 and M_3 to get one in M_2, which shows that M_2 ∈_a+b. We now prove the reverse inequality. First suppose that F ⊂ G ⊂ M satisfy G/F ≅ L, and let F'=F ∩ M_1 and F”=(F+M_1)/M_1, and similarly define G' and G”. Then either F'/G' ≅ L or F”/G”≅ L. Now suppose M ∈_n, and let F_1 ⊂ G_1 ⊂⋯⊂ F_n ⊂ G_n ⊂ M be the given chain. Define F_i', F_i”, G'_i, and G”_i as before. Thus, for each i, we have F'_i/G'_i ≅ L or F”_i/G”_i ≅ L. This shows that M_1 ∈_a and M_3 ∈_b for some a and b summing to n, which completes the proof.

(b) This follows immediately from (a).

(c) From (b), we have μ_L(M) ≥μ_L(M_i) for each i, and so μ_L(M) ≥sup_i ∈ Iμ_L(M_i). We now prove the reverse inequality. Thus suppose that μ_L(M) ≥ n, and let us show that μ_L(M_i) ≥ n for some i.

First suppose F ⊂ G ⊂ M satisfy G/F ≅ L. Let F^(i) = F ∩ M_i and G^(i) = G ∩ M_i. Since (AB5) holds, we have ⋃_i ∈ I F^(i) = F, and similarly for G. For each i, we have an injection F^(i)/G^(i)→ F/G ≅ L, and so F^(i)/G^(i) is either 0 or isomorphic to L, since L is simple. Since (AB5) holds, the direct limit of the F^(i)/G^(i) is F/G ≅ L, and so F^(i)/G^(i) cannot vanish for all i, that is, there exists some i for which it is L. Note that for i ≤ j we have an injection F^(i)/G^(i)→ F^(j)/G^(j), so if the former is non-zero so is the latter.

Now, returning to the main point, let F_1 ⊂ G_1 ⊂⋯⊂ F_n ⊂ G_n ⊂ M be the given chain in M. Let i_1, …, i_n ∈ I be such that F^(i_k)_k/G^(i_k)_k ≅ L for each k, and pick i ∈ I such that i_1, …, i_n ≤ i. Then F^(i)_k/G^(i)_k ≅ L for each k. Thus the chian F^(i)_1 ⊂ G^(i)_1 ⊂⋯⊂ M_i shows that M_i ∈_n, which completes the proof.



Suppose  is a locally noetherian Grothendieck abelian category, and that μ_L(M)=0 for all simple objects L. Then M=0.



Let M be a non-zero object of . Since  is locally noetherian, M has a non-zero noetherian submodule M'. Since M' is noetherian, it has a maximal proper submodule M”. Thus L=M'/M” is a simple subquotient of M, and so μ_L(M)>0.














 
Let  be a -linear abelian category, with  a field. Suppose that L is a simple object of  with (L)= and I is its injective envelope. Then for any M ∈ we have μ_L(M)=(M, I).



Put μ=μ_L and let ν be defined by ν(M)=(M, I). We must show μ=ν. To begin with, we show that they are equal when μ is finite, by induction on μ.

First, suppose that μ(M)=0 and let us show that ν(M)=0. Suppose f  M → I is a non-zero map. Then f(M) is a non-zero subobject of I, and thus meets L, since I is an essential extension of L, and thus contains L, since L is simple. This shows that L is a subquotient of M, which contradicts μ(M)=0. Thus there are no non-zero maps M → I and so ν(M)=0.

Next, suppose that μ(M)=ν(M) whenever μ(M)<n, and let us show that the equality continues to hold when μ(M)=n. Thus let M be given with μ(M)=n. We have already handled the n=0 case, so we can assume n>0. Thus L occurs as  a subquotient of M, say L=G/F with F ⊂ G ⊂ M. We have μ(M)=μ(F)+μ(G/F)+μ(M/G). Since μ(G/F)=1, it follows that μ(F) and μ(M/G) are both strictly less than n. Thus, by the inductive hypothesis, we have μ(F)=ν(F) and μ(M/G)=ν(M/G). Furthermore, the hypothesis (L)= ensures that (L, I) is one-dimensional, and so ν(G/F)=ν(L)=1. As ν(M)=ν(F)+ν(G/F)+ν(M/G), we thus find ν(M)=μ(M).

We have thus shown that μ(M)=ν(M) whenever μ(M) is finite. To conclude, let us show that the equality continues to hold when μ(M) is infinite. Let n>0 be an integer. Then, by the definition of μ, we can find F_1 ⊂ G_1 ⊂⋯⊂ F_n ⊂ G_n ⊂ M with G_i/F_i ≅ L for each i. Let f_i  M → I be a map extending the map G_i → G_i/F_i ≅ L ⊂ I; such an extension exists since I is injective. The f_i are linearly independent: indeed, we cannot express f_i as a linear combination of f_i+1, …, f_n since f_i(G_i) is non-zero but f_j(G_i)=0 for j>i. We thus see that ν(M)>n, which completes the proof.












 §.§ Derived functors and limits and colimits


 
Let F → be a left-exact functor of abelian categories. Suppose that:


  *  is a Grothendieck category.

  * F commutes with filtered colimits.

  * A filtered colimit of injective objects in  is F-acyclic.

Then ^i F commutes with filtered colimits.



Let {M_i}_i ∈ I be a filtered system in . Let {M_i → J_i^∙}_i ∈ I be a filtered system of injective resolutions; such a system exists since  has functorial injective resolutions 079H. We thus see that M_i → J_i^∙ is an F-acyclic resolution. The result now follows easily.


 
Let F → be an additive functor of abelian categories. Assume


  *  is a Grothendieck category.

  *  has exact countable products.

  * F commutes with countable products.

Then F () →() commutes with derived limits.



See 08U1.




 §.§ Localizing subcategories
 

Let  be a Grothendieck abelian category and let S be some collection of objects in . Recall that a localizing subcategory of  is a Serre subcategory closed under arbitrary direct sums (or, equivalently, direct limits). Let  be the smallest localizing subcategory of  containing S; we call this the localizing subcategory generated by S. The following proposition gives a concrete description of the finitely generated objects in this category under a noetherian hypothesis:

 
Suppose that  is locally noetherian. Then a finitely generated object M of  belongs to  if and only if there is a finite length filtration 0=F_0 ⊂⋯⊂ F^n=M and objects E_1, …, E_n of S such that F^i/F^i-1 is isomorphic to a subquotient of E_i for all 1 ≤ i ≤ n.



Let _0 be the subcategory of  consisting of finitely generated objects M admitting a filtration as in the statement of the proposition. Then _0 is closed under subquotients. Indeed, suppose M belongs to _0 and N is a subobject of M. Let F^0 ⊂⋯⊂ F^n and E_1, …, E_n be the given data on M, and let G^∙ = F^∙∩ N. Then G^i/G^i-1 is naturally a submodule of F^i/F^i-1, and thus isomorphic to a subquotient of E_i. This shows that N belongs to _0. The proof for quotients is similar. It is clear that _0 is closed under extensions. Thus _0 is a Serre subcategory of .

Let  be the subcategory of  consisting of all objects for which every finitely generated subobject belongs to _0. We show that  is a localizing subcategory:


  * Suppose that M belongs to  and N is a subobject of M. Since a finitely generated subobject of N is also one of M, and thus belongs to _0, we see that N belongs to .

  * Next, say that M belongs to  and N is a quotient of M. Let N_0 be a finitely generated subobject of N, and let {M_i}_i ∈ I be the collection of all finitely generated subobjects of M. Let M_i be the image of M in N. Then the image of M in N is contained in ∑_i ∈ IM_i, and so ∑_i ∈ IM_i=N. Thus, since  is a Grothendieck category, we have ∑_i ∈ I (N_0 ∩M_i)=N_0. Since N_0 is finitely generated, we have N_0=M_i ∩ N_0 for some i, and so N_0 ⊂M_i. Since M_i belongs to _0 (since M belongs to ) and N_0 is a subquotient of M_i, it too belongs to _0 (since _0 is a Serre subcategory). Thus N belongs to .

  * Suppose now that M belongs to  and there is a subobject N of M such that N and M/N belong to . Let M_0 be a finitely generated subobject of M. Let N_0=N ∩ M_0. This is finitely generated, since M_0 is finitely generated and  is locally noetherian, and thus belongs to _0, since N belongs to . Similarly, M_0/N_0 is finitely generated, being a quotient of M_0, and is a subobject of M/N, and thus belongs to _0, since M/N belongs to . Thus M_0 is an extension of objects in _0 and thus belongs to _0 since _0 is a Serre subcategory.

  * We have thus shown that  is Serre subcategory. It remains to show that it is closed under direct sums. Thus suppose that {M_i}_i ∈ I is a collection of objects in , and consider M=⊕_i ∈ I M_i. Let N be a finitely generated subobject of M. For a finite subset J of I, let M_J=⊕_i ∈ J M_i, so that M=∑_J M_J. Thus N=∑_J (M_J ∩ N). Since N is finitely generated, we have N=M_J ∩ N for some J, i.e., N ⊂ M_J. Since M_J is a finite sum of objects in , it belongs to  (since  is a Serre subcategory). Thus N belongs to _0. This shows that M belongs to , verifying the claim.

Since  is a localizing subcategory and contains S, we have ⊂ by definition of . On the other hand, it is clear that _0 ⊂, and so ⊂ since  is closed under direct limits. Thus =. If M is a finitely generated object in this category then it necessarily belongs to _0, and thus admits the requisite filtration.


The following proposition gives a simple criterion for a functor to map one localizing category into another.

 
Let →' be a cocontinuous functor of Grothendieck abelian categories, with  locally noetherian. Let S be a collection of objects in  generating a localizing subcategory , and let ' be a localizing subcategory of '. Suppose  is exact on  and (S) ⊂'. Then () ⊂'.



Let M be a finitely generated object of . By Proposition <ref>, we have a filtration 0=F^0 ⊂⋯⊂ F^n=M and objects E_1, …, E_n of S such that F^i/F^i-1 is a subquotient of E_i. By hypothesis, (E_i) belongs to '. Since  is exact on , we see that (F^i/F^i-1) is a subquotient of (E_i), and thus belongs to '. The exactness of  also shows that (M) is a succsesive extension of the objects (F^i/F^i-1), and therefore belongs to .

Now suppose that M is an arbitrary object of . Then M=_i ∈ I M_i for some direct set I, where the M_i are finitely generated subobjects of M. Since  is cocontinuous, we have (M) = _i ∈ I(M_i). By the previous paragraph, (M_i) belongs to '. Since ' is closed under direct limits, we see that (M) also belongs to '.




 §.§ Defining functors on projectives


 
Let  and  be abelian categories. Suppose that  has enough projectives, and let  be the full subcategory spanned by the projective objects. Then the restriction functor

Φ{ right exact functors → } →{ additive functors → }

is an equivalence. Moreover, if  and  are cocomplete and direct sums are exact then Φ induces an equivalence

{ cocontinuous functors → } →{ functors → commuting with all direct sums }.




Suppose that G → is an additive functor. Let M ∈, and choose a presentation P_1 → P_0 → M → 0 with P_1 and P_0 projective. Define F(M) to be the cokernel of the map G(P_1) → G(P_0). One easily sees that this yields a well-defined right exact functor F →. Define Ψ(G)=F. One easily sees that Ψ defines a quasi-inverse functor to Φ. This yields the first equivalence. For the second, simply note that if F and G correspond under the first equivalence then F is cocontinuous if and only if G commutes with all direct sums.


 
Let  and  be Grothendieck abelian categories. Let  be a full subcategory of  whose objects are finitely generated projectives. Suppose that every object of  is a quotient of a sum of objects of . Then the restriction functor

{ cocontinuous functors → } →{ additive functors → }

is an equivalence.



Let ⊂ be the full subcategory spanned by the projective objects. Appealing to the previous proposition, it suffices to show that the restriction functor

{ functors → commuting with all direct sums } →{ additive functors → }

is an equivalence.

Let ' be the following category. An object is a family {Q_i}_i ∈ I with Q_i ∈. A morphism {Q_i}_i ∈ I→{Q'_j}_j ∈ J consists of giving, for each i, a morphism Q_i →⊕_j∈ J_i Q'_j, where J_i is a finite subset of J. Let ” be the Karoubian envelope of '. There is a natural functor ' → taking {Q_i}_i ∈ I to ⊕_i ∈ I Q_i, which extends to a functor ”→ since  is Karoubian. This functor is an equivalence: the key point is that if Q ∈ then any morphism Q →⊕_i ∈ I M_i with M_i ∈ factors through some finite direct sum, since Q is finitely generated.

Now, any functor → induces a functor ”→, since ” is obtained naturally from  and  has all direct sums, and thus a functor →. One verifies that this construction provides the requisite quasi-inverse to the restriction functor.




 §.§ An acyclicity criterion


 
Let  and  be Grothendieck abelian categories, and let () →() be a functor. Suppose that:


  *  is exact and cocontinuous.

  *  takes cones to cones, that is, if f  M → N is a morphism in () then we have a canonical isomorphism ((f)) ≅((f)) in ().

  *  “only adds terms to the left,” that is, (^≤ n()) ⊂^≤ n(), where ^≤ n is the category of chain complexes supported in degrees ≤ n.

Then  takes acyclic complexes to acyclic complexes and quasi-isomorphisms to quasi-isomorphisms.



In light of condition (b), the two conclusions of the proposition are equivalent. We prove the statement about acyclic complexes. Thus let M be an ayclic complex in (). To show that (M) is acyclic, we proceed in four steps:

Step 1: M is a 2-term complex. In this case, M=(f) for some isomorphism f of 1-term complexes. Thus (M) ≅((f)) by condition (b), and therefore acyclic, since (f) is an isomorphism.

Step 2: M is a bounded complex. For a complex N, let t_≤ n(N) denote its canonical truncation: the terms are given by

t_≤n(N)^k = 
N^k     if k<n 

(d N^n →N^n+1)     if k=n 

0     if k>n 

Then t_≤ n(N) is a subcomplex of N, and is acyclic if N is. Suppose now that M^k=0 for k>n. Consider the exact sequence of complexes

0 →t_≤n-1(M) →M →M' →0.

Then M' is a 2-term acyclic complex, while t_≤ n-1(M) is an acyclic complex that is shorter than M. By Step 1, (M') is acyclic, while by induction on the length of the complex, (t_≤ n-1(M)) is acyclic. By condition (a), we see that

0 →(t_≤n-1(M)) →(M) →(M') →0

is exact, and so (M) is acyclic.

Step 3: M is bounded above. Let n be given, and let us show that ^n((M))=0. Consider the exact sequence

0 →t_≤n-1(M) →M →M' →0.

All the complexes above are acyclic, and M' is bounded. By Step 2, (M') is acyclic. We thus see that the natural map

^n((t_≤n-1(M))) →^n((M))

is an isomorphism. But t_≤ n-1(M)^k=0 for k ≥ n, so (t_≤ n-1(M))^k=0 for k ≥ n as well by condition (c). Thus the above homology groups vanish.

Step 4: the general case. The complex M is the direct limit of the acyclic subcomplexes t_≤ n(M), each of which is bounded above. By condition (a), we thus see that (M) is the direct limit of the complexes (t_≤ n(M)), each of which is acyclic by Step 3. Thus (M) is acyclic.



 
Let  and  be Grothendieck abelian categories, let ,() →() be functors, and let ϕ→ be a natural transformation. Suppose that:


  *  and  satisfy the three conditions of Proposition <ref>.

  *  has enough projectives.

  * If P ∈ is projective then ϕ_P (P) →(P) is a quasi-isomorphism.

Then ϕ_M (M) →(M) is a quasi-isomorphism for all M ∈().



Suppose now that M is a bounded complex of projective modules. We show that (M) →(M) is a quasi-isomorphism by induction on the length of the complex. Assuming M is non-zero, let n be maximal such that M^n is non-zero, and let N be the “stupid truncation” of M to degrees <n; that is, N^k=M^k for k<n and N^k=0 for k ≥ n. We then have an exact sequence of complexes

0 →N →M →K →0,

where K=M^n[-n] is concentrated in degree n. This gives rise to a diagram


0 [r]     (N) [r] [d]     (M) [r] [d]     (K) [r] [d]     0 

0 [r]     (N) [r]     (M) [r]     (K) [r]     0 

The rows are exact since  and  are exact. The right vertical map is a quasi-isomorphism by assumption, while the left one is by the inductive hypothesis. Thus the middle one is too.

Now suppose that M is a bounded above complex of projective modules. We show that ^i((M)) →^i((M)) is an isomorphism for all i. Thus let i be given. Let N be the stupid truncation of M to degrees ≤ i-1, and let K be the quotient M/N. We again get a diagram like the above. Since K is bounded, we know that ^i((K)) →^i((K)) is an isomorphism. Since N is supported in degrees ≤ i-1, so is the complex (N), and so ^i((K))=^i+1((K))=0; similarly for the  versions. It follows that ^i((M)) →^i((M)) is an isomorphism.

Suppose now that M is an arbitrary bounded above complex. Let P → M be a quasi-isomorphism, with P a bounded above complex of projectives. By Proposition <ref>,  and  takes quasi-isomorphisms to quasi-isomorphisms. Thus (P) →(M) is a quasi-isomorphism, and similarly for . We have shown that (P) →(P) is a quasi-isomorphis. Considering the obvious commutative square, it follows that (M) →(M) is a quasi-isomorphism.

Finally, suppose that M is an arbitrary complex. Then M is the direct limit of bounded above subcomplexes. Since  and  commute with direct limits, the result follows.




 §.§ An equivalence criterion


For an abelian category , let (), resp. (), denote the full subcategory spanned by the injective, resp. indecomposable injective, objects of .

 
Let → be a functor of locally noetherian Grothendieck abelian categories. Suppose that:


  *  is left-exact and commutes with filtered colimits.

  *  takes finitely generated objects to finitely generated objects.

  *  induces an equivalence () →().

Then  is an equivalence.



We first show that  is fully faithful. To this end, we say that a pair (M,N) of objects of  is good if the  natural map

_(M, N) →_((M), (N))

is an isomorphism. From (c), we see that (I, J) is good for all I,J ∈(). We show that all pairs of objects are good in several steps.

Step 1.
Suppose that I ∈() and J ∈(). Write I=⊕_α∈ I_α where  is an index set and each I_α is an indecomposable injective; this is possible because  is locally noetherian. We have a commutative square


_(I, J) [r] @=[d]    
_((I), (J)) @=[d] 

∏_α∈ _(I_α, J) [r]    
∏_α∈ _((I_α), (J)) 

To get the right vertical equiality, we used that (I)=⊕_α∈(I_α), which follows from (a). Since (I_α, J) is good for all α, the bottom map is an isomorphism. Thus the top map is an isomorphism as well. We conclude that (I, J) is good for all I ∈() and J ∈().

Step 2.
Now suppose that J ∈() and M ∈ is arbitrary. Choose an exact sequence

0 →M →I_0 →I_1

with I_0, I_1 ∈(). This is possible because  has enough injectives. We then get a diagram


_(I_1, J) [r] [d]     _(I_0, J) [r] [d]     _(M, J) [r] [d]     0 

_((I_1), (J)) [r]     _((I_0), (J)) [r]     _((M), (J)) [r]     0 

The rows are exact: this follows from the fact that  is left-exact and that J and (J) are injective objects. Since (I_0, J) and (I_1, J) are good (Step 1), the two left vertical maps are isomorphisms, and so the right vertical map is as well. We conclude that (M, J) is good for all M ∈ and all J ∈().

Step 3.
Now suppose that J ∈() and M ∈^. Write J=⊕_α∈ J_α with J_α indecomposable. We have a commutative square


_(M, J) [r] @=[d]    
_((M), (J)) @=[d] 

⊕_α∈ _(M, J_α) [r]    
⊕_α∈ _((M), (J_α)) 

The left vertical equality follows from M being finitely generated (and thus noetherian). The right vertical equality follows from (M) being finitely generated, which comes from (b), and the identification (J)=⊕_α∈(J_α), which comes from (a). Since (M, J_α) is good (Step 2), the bottom map is an isomorphism. Thus the top one is as well. We conclude that (M, J) is good for all J ∈() and all M ∈^.

Step 4.
Let J ∈() and M ∈. Write M=_α∈ M_α where  is a directed set and each M_α is a finitely generated subobject of M. This is possible since  is locally noetherian. We have a commutative square


_(M, J) [r] @=[d]    
_((M), (J)) @=[d] 

_(M_α, J) [r]    
_((M_α), (J)) 

Here we are simply using the universal property of direct limits, and the fact that (M)=(M_α), which comes from (a). Since each (M_α, J) is good (Step 3), the lower map is an isomorphism. Thus the upper map is an isomorphism. We conclude that (M, J) is good for all M ∈ and all J ∈().

Step 5.
Finally, let M,N ∈ be arbitrary. Pick an exact sequence

0 →N →J_0 →J_1

with J_0, J_1 ∈(). We obtain a commutative diagram


0 [r]     _(M, N) [r] [d]     _(M, J_0) [r] [d]     _(M, J_1) [d] 

0 [r]     _((M), (N)) [r]     _((M), (J_0)) [r]     _((M), (J_1)) 

The rows are exact: this follows from the left exactness of  and . Since (M, J_0) and (M, J_1) are good (Step 4), the right two vertical maps are isomorphisms. Thus the left vertical map is an isomorphism. We conclude that (M, N) is good for all M,N ∈.

Completion of proof.
We have shown that  is fully faithful. To complete the proof, we must show that  is essentially surjective. We first claim that given J ∈() there exists I ∈() with (I) ≅ J. Indeed, write J=⊕_α∈ J_α with J_α∈(). By (c), for each α we can find some I_α∈() and an isomorphism J_α≅(I_α). Let I=⊕_α∈ I_α, which is an injective object of . Then (I)=⊕_α∈(I_α) ≅ J, where in the first identification we have used (a). This establishes the claim.

Now let N ∈ be an arbitrary object. Choose an exact sequence

0 →N →J_0 g→ J_1

with J_0, J_1 ∈(). By the previous paragraph, we can find I_0, I_1 ∈() with (I_0) ≅ J_0 and (I_1) ≅ J_1. Since  is fully faithful, the morphism g ∈_(J_0, J_1) comes from a morphism f ∈_(I_0, I_1). Let M=(f). Then

(M) ≅((f)) ≅((f)) ≅(g) ≅N,

where in the second identification we used the left-exactness of . Thus  is essentially surjective, which completes the proof.




 §.§ Exact sequences of functors


 
Let  and  be Grothendieck abelian categories. Let

0 →F_1 →F_2 →F_3 →0

be an exact sequence in the functor category (, ).


  * If two of the functors are exact then so is the third.

  * If two of the functors commute with filtered colimits then so does the third.




(a) This is a simple consequence of the 9-lemma (also called the 3 × 3 lemma).

(b) Suppose that {M_i}_i ∈ is a filtered system. Consider the diagram


0 [r]     F_1(M_i) [r] [d]     F_2(M_i) [r] [d]     F_3(M_i) [r] [d]     0 

0 [r]     F_1(M_i) [r]     F_2(M_i) [r]     F_3(M_i) [r]     0 

The vertical maps are the canonical maps. The rows are exact since  and  are Grothendieck categories. Thus if two of the vertical maps are isomorphisms then so is the third.




 §.§ Injectives


The following is a version of Baer's criterion:

 
Let  be a Grothendieck abelian category. Suppose that every object is the union of its finitely generated subobjects; for example,  could be locally noetherian. Suppose that I is an object of  satisfying the following condition: given any diagram


    I 

M [r]^i [ru]^f     N @..>[u]_g 

in which M and N are finitely generated and i is injective, one can find a morphism g such that the diagram commutes. Then I is injective.



The proof is similar to the usual proof of Baer's criterion; see 079G for a similar result.



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