Abstract Abstract. We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including Q and F p , we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals. Introduction Although the theory of (mixed) motives has in recent years made significant progress, especially in regards to motivic sheaves (i.e. motives over general schemes), our understanding of motives over a field is still limited. It is therefore natural to restrict one's attention to certain subclasses of motives in order to gain some intuition. One subclass which has proven particularly fruitful for that purpose is the class of (mixed) Tate motives. These are the motives which can be constructed from the simple building blocks Z(n) (the "Tate twists", n ∈ Z) by extensions, (de)suspensions, tensor products, and direct summands. Another view is that they encode the motivic cohomology of the base field. As should be clear from these two descriptions, Tate motives are at the same time relatively simple (compared to the class of all motives), and yet contain complex and interesting information. A striking illustration of the latter fact is their role in the theory of periods, particularly their relation to multiple zeta values, as explained in [11; 16; 37] . Introduction One long strand of research into Tate motives has been concerned with uncovering the structural properties of the category of Tate motives, with some success [e.g. 7; 9; 23; 27; 28; 35; 42] . The motivation for the present paper is to advance in this direction, by focusing on the particular properties visible to tensor triangular geometry. Indeed, Tate motives form a triangulated category with a compatible tensor structure (a tt-category for short), and one may try to classify its tt-ideals, i.e. its thick subcategories closed under tensor products with arbitrary Tate motives. The classification of tt-ideals gives insights into the composition and complexity of the tt-category; if two objects generate different tt-ideals, this means that they cannot be constructed out of each other using extensions, (de)suspensions, tensor products, and direct summands: they are quite far from being "the same". Introduction The main device in tt-geometry is Balmer's tt-spectrum [3] , a spectral space associated to the tt-category which encodes the tt-ideals through its topology. For a number of classical mathematical domains, the tt-spectrum has been studied; we refer to [5] for an overview of the basic theory, its early successes and applications. In relation to the present article, there has been earlier work on the tt-spectrum of certain motives (including Tate motives) [26; 33] . However, the arguments were restricted to rational coefficients (and certain base fields) so that the tt-spectrum turned out to be a singleton space. In other words, in these categories every object can be constructed from every other (non-zero) object. Introduction In contrast, here we work with integral coefficients and find non-trivial ttspectra. For example, we determine completely the tt-spectrum of Tate motives over the algebraic numbers (for more general base fields see Theorem 8.6 and Corollary 9.3): Theorem 1. The tt-spectrum of DTM gm (Q, Z) consists of the following points, with specialization relations, depicted by the lines, going upward. Here, runs through all prime numbers, and the points are defined by the vanishing of the cohomology theories as indicated on the right. Moreover, the proper closed subsets are precisely the finite subsets stable under specialization. Introduction Given that motives are supposed to encode the cohomological aspects of algebraic varieties, it is of course not surprising that the tt-spectrum above contains points coming from the different cohomology theories available. But it is reassuring to find that all points are of this form, lending some support to the belief that motives are the universal cohomology theory. (Having said that, it would of course be very interesting to find non-expected points as these could hint at cohomology theories not yet discovered, or other more mysterious phenomena.) We also mention that some interesting properties of Tate motives are encoded in the tt-spectrum in a slightly subtle way. For example, that the topology is as coarse translates to the fact that the motivic cohomology of (Tate) motives behaves in a certain sense like a finitely generated abelian group (see Remark 5.7). Of course, from Theorem 1 we easily deduce a classification of the tt-ideals in DTM gm (Q, Z) (see Theorem 9.5) . Introduction In order to generalize from rational to integral coefficients we need to understand the case of finite coefficients. The following is our result in that direction (cf. Corollary 8.3). Introduction Theorem 2. Let F be an algebraically closed field, and a prime number invertible in F. The tt-spectrum of DTM gm (F, Z/ ) is canonically isomorphic to the homogeneous spectrum of the polynomial ring in one variable Z/ [β]. Introduction In the proof of this theorem we use on the one hand Positselski's description of Tate motives in terms of filtered Galois representations [35] , and on the other hand our study of the tt-geometry of filtered modules in [18] . Some of Balmer's new results in tt-geometry [6] allow us then to patch our findings together with the findings for rational coefficients by Peter [33] , to arrive at the integral statement of Theorem 1. Introduction The theory of étale motives is closely related to the theory of motives -for example the two theories coincide with rational coefficients. We study étale motives both for their own interest, and because we will use the comparison of the two theories to bear on our understanding of Tate motives. This works because with torsion coefficients the étale theory is much simpler: the Rigidity Theorem of Suslin and Voevodsky identifies étale motives with Galois representations. We will use this result together with Rost, Voevodsky and others' resolution of the Bloch-Kato conjecture to confirm this simplicity in tt-geometric terms (cf. Theorem 6.2). Introduction Theorem 3. Let F be a field, and a prime number invertible in F. Assume that F contains a primitive th root of unity (respectively, primitive 4th root of unity if = 2). Then the tt-spectrum of DM´e t gm (F, Z/ ) has a single point. Note that this statement is about the triangulated category of all étale motives. The analogous statement about étale Tate motives is an immediate consequence. Introduction When the tt-category is rigid, i.e. every object has a strong tensor dual, then the tt-spectrum can be endowed with a natural structure sheaf turning it into a locally ringed space. This has been used by Balmer to, for example, recover a topologically noetherian scheme from its category of perfect complexes [3: 6.3] . Over fields such as the algebraic numbers, where we know the underlying topological space of the tt-spectrum of Tate motives completely by Theorem 1, we will describe this sheaf explicitly (cf. Proposition 10.2). To do so we need to understand the category one obtains from Tate motives by inverting the image of the Bott elements β : Z/ (0) → Z/ (1) appearing in Theorem 2 (it amounts to the choice of a primitive th root of unity). It has been shown by Haesemeyer and Hornbostel in [22] that (under some assumptions on the base field), by inverting motives with finite coefficients with respect to the Bott element one obtains étale motives. We will upgrade this result to integral coefficients (cf. Theorem C.4). Theorem 4. Let F be a field of exponential characteristic p, containing all roots of unity of order coprime to p, and of finite -cohomological dimension for all primes p. Then there are canonical equivalences of tt-categories Theorem 4. The same result holds for the effective versions. Acknowledgment I would like to thank Paul Balmer, Shane Kelly, and Simon Pepin Lehalleur for their valuable input on an earlier version of this article. I'm also grateful to Joseph Ayoub for an encouraging discussion concerning Appendix C. Conventions Our conventions regarding tensor triangular geometry mostly follow those of [3; 4] . A tensor triangulated category (or tt-category for short) is a triangulated category with a compatible (symmetric, unital) tensor structure. If not specified otherwise, the tensor product is denoted by ⊗ and the unit by 1. A tt-functor is an exact tensor functor between tt-categories. Conventions A tt-ideal in a tt-category T is a thick subcategory I ⊂ T such that T ⊗ I ⊂ I. If S is a set of objects in T we denote by S the tt-ideal generated by S. To a small tt-category T one associates a ringed space Spec(T ), called the tt-spectrum of T , whose underlying topological space is denoted by Spc(T ). It is a spectral space and consists of prime ideals in T , i.e. tt-ideals I such that a ⊗ b ∈ I implies a ∈ I or b ∈ I. A base for the closed subsets of the topology is given by the supports supp(a) of objects a ∈ T ; here, supp(a) = {P | a P}. The complement of supp(a) is denoted by U(a). If T is rigid then Spec(T ) is a locally ringed space. 1 All rings are commutative with unit, and morphisms of rings are unital. For R a ring, we denote by Spec(R) the Zariski spectrum of R (considered as a locally ringed space) whereas Spc(R) denotes its underlying topological space (as for the ttspectrum). We adopt similar conventions regarding graded rings R: they are commutative in a general graded sense [4: 3.4] , and possess a unit. Spec h (R) denotes the homogeneous (Zariski) spectrum with underlying topological space Spc h (R). This differs from Proj(R) in that the prime ideals making up the underlying space may contain the irrelevant ideal in R. It is still a spectral space though. Conventions Recall also that Balmer constructs [4] comparison maps between the tt-spectrum and certain Zariski spectra. Explicitly, there is a canonical spectral morphism ρ : Spec(T ) → Spec(R T ), where R T = End T (1) denotes the endomorphism ring of the unit in T , called the central ring. More generally, fixing an invertible object u ∈ T , he considers the graded central ring R Conventions T homogeneous | cone(r) P}. The map ρ is just the restriction of ρ Conventions • to the degree 0 part. We will repeatedly use the fact that ρ (respectively ρ • ) is a homeomorphisms if and only if it is an isomorphism of locally ringed spaces [4: 6.11]. Conventions 3. Triangulated category of (Tate) motives Fix a ring R and a field F. In these two preliminary sections (Section 3 and 4) we are going to recall some generalities on categories of motives over F with coefficients in R. Many of the constructions and proofs go back to work of Voevodsky, Suslin, Bloch, Levine, and many others. Our discussion will be too brief for somewe recommend [30; 40] as introductions instead. As main reference for this section we will use [12] . Conventions There is a (large) tt-category of "big" motives DM(F, R) [12: 11.1.1, 11.1.2] constructed from the derived category of Nisnevich sheaves with transfers of Rmodules on the category Sm/F of smooth (finite type, separated) F-schemes. In particular, it comes with an "associated motive" functor Conventions The tensor structure on DM(F, R) is determined by two facts: Conventions • The functor R(−) is symmetric unital monoidal (also with respect to transfers), for example Conventions , and R(Spec(F)) =: R(0) is the unit. (If the context allows we will write simply R for R(0).) • The reduced motive of P , denoted by R(1) [2] , is (⊗-)invertible. One then gets for any integer n an invertible R(n), called the Tate twist of weight n. Clearly, R(i) ⊗ R(j) = R(i + j), and the dual of R(i) is R(−i). Conventions The triangulated category DM(F, R) is compactly generated, and a set of compact generators is given by the motives of smooth F-schemes [12: 11.1.6]. In fact, in a sense one can make precise, DM(F, R) is generated by R(X) (X runs through smooth F-schemes) and R(−1), subject to Nisnevich descent and the relations R(1) ⊗ R(−1) = R(0), R(A 1 F ) = R(0). We denote the subcategory of compact objects by DM gm (F, R). Its objects are often called geometric motives, or just motives if no confusion with big motives is possible (or if the distinction in the given context is immaterial). By what was said above, DM gm (F, R) is the thick subcategory generated by R(X)(n) := R(X) ⊗ R(n), where X is a smooth F-scheme and n ∈ Z. It is a (small) idempotent complete tt-category. Moreover, it is rigid if the exponential characteristic 2 of F is invertible in R [25: 5.5.14; 13: 8.1]. The triangulated category of Tate motives is the thick subcategory generated by the Tate twists R(n), n ∈ Z. It is denoted by DTM gm (F, R). It is a (small) rigid, idempotent complete tt-category. There is also a "big" version: DTM(F, R) denotes the localizing subcategory of DM(F, R) generated by Tate twists. It is a (large) tt-category. Conventions Given a ring morphism R → R , one can associate to a Nisnevich sheaf with transfers of R-modules F the sheafification of X → F(X) ⊗ R R , a Nisnevich sheaf with transfers of R -modules. This induces an adjunction Conventions the right adjoint being induced by forgetting the R -structure. The functor γ * is conservative [see the proof of 2: A.6 or 14: 5.4.2], and for any motive M ∈ DM(F, R), we have γ * γ * (M) = M ⊗ R , where R is the constant sheaf associated to R , considered as an object in DM(F, R). The tt-functor γ * sends R(X)(n) to R (X)(n) and therefore restricts to tt-functors Conventions If R is a perfect R-module (i.e. R ∈ D perf (R)) then the right adjoint γ * also preserves compact objects. Conventions The hom sets in the triangulated category of motives are closely related to important algebraic geometric invariants. Specifically, for a smooth F-scheme X and integers m, n, the groups Conventions ) are the motivic cohomology groups of X with coefficients in R. There is a canonical identification [41] , Conventions with a direct generalization of Chow groups, called Bloch's higher Chow groups, thereby linking motivic cohomology to algebraic cycles. In particular, the motivic cohomology groups of Spec(F) on the "diagonal" are canonically isomorphic to Milnor K-theory [32; 39] : Conventions More generally, we define for any motive M ∈ DM(F, R) and any ring R over R, Conventions the motivic cohomology groups of M with coefficients in R . Triangulated category ofétale (Tate) motives We want to discuss étale versions of the constructions in the previous section. Thus instead of Nisnevich sheaves with transfers we consider étale sheaves with transfers. Our main references for this section are [1; 2; 14] . Triangulated category ofétale (Tate) motives There is a (large) tt-category of "big" étale motives DM´e t (F, R) [1: §4.1.1; 14: 2.2.4] constructed from the derived category of étale sheaves with transfers of R-modules on the category Sm/F of smooth (finite type, separated) F-schemes. In particular, it comes again with a symmetric unital monoidal "associated étale motive" functor Triangulated category ofétale (Tate) motives In contrast to the situation of the previous section, the image is in general not compact, and DM´e t (F, R) is in general not compactly generated (the reason being that the étale cohomological dimension of F can be infinite). We denote the thick subcategory generated by R´e t (X)(n) for X smooth and n an integer by DM´e Triangulated category ofétale (Tate) motives to the subcategory of D(G F , Z/ ) spanned by complexes with bounded, finite dimensional cohomology. This is a tt-functor which moreover factors through a fully faithful tt-functor [14: 5.5.4] , abusively denoted by the same symbol, (4.2) Re : DM´e Torsion and finite generation in motivic cohomology In this section we collect some basic properties concerning rational and modmotivic cohomology for (Tate) motives. The discussion will culminate in the proof that the motivic cohomology groups of (Tate) motives behave "as if they were finitely generated". We refer to Remark 5.7 for elaboration. Torsion and finite generation in motivic cohomology For the reader unfamiliar with the objects introduced in the previous two sections, this is a good opportunity to get better acquainted with the formalism. Throughout the section, F is an arbitrary field. Torsion and finite generation in motivic cohomology We start with a result characteristic of Tate motives. Torsion and finite generation in motivic cohomology Lemma 5.1. Let M ∈ DTM gm (F, R) be a Tate motive. The following are equivalent: Torsion and finite generation in motivic cohomology Since the set of Tate twists generates DTM gm (F, R) we see that M ∨ = 0 and this implies M = 0. Convention 5.2. From now on and until the end of the section we assume that R is a localization of Z such that Z ⊂ R ⊂ Q. Convention 5.2. Let F : DTM gm (F, R) → T be a tt-functor, and M ∈ DTM gm (F, R) a Tate motive. We say that M is F-acyclic if F(M) = 0. The F-acyclic objects clearly define a tt-ideal in DTM gm (F, R). If A ∈ DM(F, R) is any motive, we say that M is A-acyclic if it is acyclic with respect to the functor −⊗A : DTM gm (F, R) → DM(F, R). We will be most interested in A = Z/ ( a prime) or A = Q. Lemma 5.4. Let S be the multiplicative subset Z\0. In each case of motives, Tate motives, étale motives, or étale Tate motives, we have canonical equivalences of tt-categories Convention 5.2. where (−) denotes the idempotent completion. Convention 5.2. Proof. The categorical (or Verdier) localization at S is the naive localization (i.e. the category obtained by localizing each hom set), by [2: 9.1; 4: 3.6]. This gives the first equivalence. For the second equivalence, the étale version is [14: 5.4.9] ; the non-étale version is simpler, and can be found in [12: 11.1.5]. Convention 5.2. Corollary 5.5. Let M ∈ DTM gm (F, R) be a Tate motive. The following are equivalent: Convention 5.2. Proof. By the change of coefficients adjunction for R → Q, we have Convention 5.2. , and by Lemma 5.1 these groups all vanish if and only if γ * M = 0. But γ * is conservative (Section 3) so this is equivalent to Convention 5.2. i.e. it is equivalent to M being Q-acyclic. Lemma 5.4 shows that the second and third condition are equivalent as well. Convention 5.2. The first part of the proof also gives the following result. Convention 5.2. Lemma 5.6. Let M ∈ DTM gm (F, R) be a Tate motive, and a prime number. The following are equivalent: Convention 5.2. Remark 5.7. Among finitely generated abelian groups there is a stark divide between finite and infinite ones. This simple observation is used in many arguments in algebraic topology which involve the universal coefficient theorem. More explicitly, if X is a topological space with finitely generated cohomology groups H Convention 5.2. • (X, Z) then there is the following dichotomy: This observation will be seen to have important ramifications for the topology of the tt-spectrum of Tate motives. Convention 5.2. Proposition 5.8. Let M ∈ DTM gm (F, R) be a Tate motive. Then the following dichotomy holds: Convention 5.2. • Convention 5.2. We have phrased most of the results in this section so far for Tate motives but it should be remarked that they equally hold for étale Tate motives with the same proofs, taking into account the following observations: Convention 5.2. • Rational étale motivic cohomology is the same thing as rational motivic cohomology as discussed above (Eq. (4.4)). Convention 5.2. • By Rigidity (Eq. (4.2)), mod-étale motivic cohomology is the same thing as mod-étale cohomology. Convention 5.2. • One sometimes has to modify the statements and arguments to take into account that the category of étale motives is Z[1/p]-linear, where p is the exponential characteristic of F. For the sake of illustration, and since we will use the étale version of Proposition 5.8 later on, we will state it explicitly here. Proposition 5.10. Let M ∈ DTM´e t gm (F, R) be an étale Tate motive. Then the following dichotomy holds: Convention 5.2. Proof of Proposition 5.8 and Proposition 5.10. We will give the proof of the first bullet point for the Nisnevich topology, and of the second bullet point for the étale topology. Convention 5.2. Assume that M ∈ DTM gm (F, R) has trivial rational motivic cohomology, i.e. M is torsion by Corollary 5.5. Let n ∈ Z >0 such that n : M → M is the zero map. Let be any prime not dividing n. We are going to show that H Convention 5.2. •,• M (M, R/ ) = 0, and this will complete the proof of the first bullet point. Consider the object γ * M ∈ DTM gm (F, R/ ), where γ * is the change of coefficients functor associated to R → R/ . Multiplication by n on γ * M is at the same time zero (since it is so on M), and an isomorphism (since the category DTM gm (F, R/ ) is R/ -linear). We conclude that γ * M = 0 as claimed, cf. Lemma 5.6. We now come to the second bullet point. Let ∈ F × \R × be a prime, M ∈ DTM´e t gm (F, R) and assume we are given a morphism M → Q(n)[m] representing a non-trivial rational motivic cohomology class. Assuming that H m,n M (M, R/ ) = 0, we will deduce a contradiction. Equivalently we are given a non-trivial morphism Convention 5.2. In other words we may assume m = n = 0 to simplify the notation. Convention 5.2. Consider the adjunction associated to the localization R → Z : Convention 5.2. Since both R/ and Q are in the image of γ * it suffices to prove the claim for γ * M. In other words, we may assume R = Z . Convention 5.2. In this case Convention 5.2. and so by compactness of M, the given cohomology class f : M → Q factors as Convention 5.2. Again, π f 1 is trivial, and f 1 factors through f 2 etc. We conclude that f 0 , and consequently f , factors through Convention 5.2. But this is impossible, as there are no non-trivial morphismsẐ → Q! (Cf. [14: 7.2.8]; the reader might find the discussion in [14: 7.2] helpful in general for analysing the category DTM´e t gm (F, Z ).) Although we are mainly interested in Tate motives in this article, it should be said that the arguments in this section easily generalize to (geometric!) motives. In particular, we have just shown that a similar dichotomy holds in DM gm (F, R) or DM´e t gm (F, R): Proposition 5.11. Let M ∈ DM gm (F, R) (or DM´e t gm (F, R)) be a motive. Then the following dichotomy holds: 3 Here and in the rest of the proof, by homotopy colimits and limits in triangulated categories we mean the "telescope construction" of [10] . That these coincide with the (∞, 1)-categorical notions has been verified in [24: App. 1]. Convention 5.2. • If M ⊗ Q = 0 then for almost all primes we have M ⊗ R/ = 0. Convention 5.2. • If M ⊗ Q 0 then for all primes we have M ⊗ R/ 0. Convention 5.2. This can again be phrased in terms of motivic cohomology, using the generators R (X)(n) for X smooth, n ∈ Z, R = Q respectively R = R/ . tt-geometry ofétale (Tate) motives The tt-geometry of étale Tate motives is easier to describe than the one of Tate motives, mostly due to our good understanding of étale motives with torsion coefficients (the Rigidity Theorem, see Eq. (4.2)). In fact, with torsion coefficients it is not more difficult to describe the tt-geometry of étale motives (not necessarily Tate) at the same time, which is what we are going to start with. The integral versions considered subsequently require the (rational) motivic cohomology of the base field to satisfy certain vanishing conditions, and hold only for Tate motives. tt-geometry ofétale (Tate) motives Fix a field F and a prime number invertible in F. We want to recall the Bloch-Kato conjecture (now a theorem). The short exact sequence of étale sheaves tt-geometry ofétale (Tate) motives (F, µ ) in étale cohomology. Using the cup product this extends to a morphism of graded rings tt-geometry ofétale (Tate) motives which clearly annihilates . Voevodsky, Rost, and others, show that the induced map is an isomorphism (as Bloch and Kato conjectured): tt-geometry ofétale (Tate) motives In order to apply this result we need to know the homogeneous spectrum of the graded rings involved. tt-geometry ofétale (Tate) motives , and so, again, we find K tt-geometry ofétale (Tate) motives Theorem 6.2. Let F be a field, and let be a prime invertible in F. Assume that F contains a primitive th root of unity (respectively, primitive 4th root if = 2). Then both canonical morphisms in the following composition are isomorphisms of locally ringed spaces: tt-geometry ofétale (Tate) motives Proof. The proof will proceed in several steps. tt-geometry ofétale (Tate) motives (1) Note that it suffices to prove that Spec(DM´e t gm (F, Z/ )) has at most one point. Since DM´e t gm (F, Z/ ) is not the trivial category, its spectrum then has exactly one point, and therefore the composition ρ : Spec(DM´e t gm (F, Z/ )) → Spec(Z/ ) is a homeomorphism. Since the inclusion DTM´e t gm (F, Z/ ) → DM´e t gm (F, Z/ ) is (fully) faithful, the induced morphism on tt-spectra is surjective [6: 1.8]. It follows that if the composition is a homeomorphism then so are both maps in the statement. But ρ being a homeomorphism already implies that it is an isomorphism of locally ringed spaces. tt-geometry ofétale (Tate) motives is an injection. Since we are assuming that Z/ µ , and by Bloch-Kato, the target of this map is the homogeneous spectrum of Milnor K-theory K M • (F)/ . But we found in Lemma 6.1 that the latter has indeed just a single point. tt-geometry ofétale (Tate) motives To complete the proof we need to compare the tt-spectrum of D b c (G, k), for a (discrete) field k and a profinite group G, to the homogeneous spectrum of the cohomology ring of G. This will be done in two steps. Consider the category mod(G, k) of finite dimensional discrete G-modules over k, and its bounded derived category D b (mod(G, k)). There is a canonical tt-functor tt-geometry ofétale (Tate) motives The functor ι of Eq. (6.3) induces an injection on tt-spectra: tt-geometry ofétale (Tate) motives Proof. By (the proof of) [15: 4.6] , the objects of D b c (G, k) are represented by bounded complexes of finite dimensional discrete G-modules over k. In other words, ι is essentially surjective. This clearly implies that the induced map on spectra is injective [3: 3.8]. tt-geometry ofétale (Tate) motives Let us now consider the graded central ring R tt-geometry ofétale (Tate) motives ) with respect to k [1] . The following statement generalizes the analogous result for finite groups which was proved in [4: 8.5 ], completing work of many others. Our proof will consist in reducing to the finite case and is therefore not independent. Proposition 6.5. Let G be a profinite group, and k a field. Then: tt-geometry ofétale (Tate) motives ( tt-geometry ofétale (Tate) motives is an isomorphism of locally ringed spaces. tt-geometry ofétale (Tate) motives Proof. The group G is the inverse limit lim ← − −i G i of a cofiltered diagram I i → G i of finite groups (with the discrete topology). For every transition map G i → G j we obtain a functor mod(G j , k) → mod(G i , k) by restricting the action. Notice that a finite dimensional G-module is discrete (i.e. the action of G is continuous) if and only if the action factors through G i , for some i. It follows easily that lim − − →i mod(G i , k) = mod(G, k) and this equivalence passes first to the level of bounded cochain complexes, and then to the bounded derived category so that we have lim tt-geometry ofétale (Tate) motives We now obtain the first part of the statement since R tt-geometry ofétale (Tate) motives For the second statement, consider the following square which is commutative by naturality of ρ tt-geometry ofétale (Tate) motives , the bottom horizontal map is a homeomorphism. Necessarily then, the left vertical map is a homeomorphism as well. We now conclude since ρ • is then automatically an isomorphism of locally ringed spaces. tt-geometry ofétale (Tate) motives Having described the tt-geometry for torsion coefficients, we next consider rational coefficients. Here, we restrict to Tate motives in order to invoke the results of [33] . Recall also (Eq. (4.4)) that we have an equivalence of tt-categories DTM gm (F, Q) DTM´e t gm (F, Q) and the discussion therefore applies to both topologies; we will phrase them for DTM gm (F, Q). tt-geometry ofétale (Tate) motives Consider H m,n M (Spec(F), Q), the rational motivic cohomology ring of the field (see Eq. (3.2)). There is the following relation between Bloch's higher Chow groups and algebraic K-theory (cf. Eq. (3.3)): tt-geometry ofétale (Tate) motives where the latter denotes the weight n eigenspace of the Adams operations (ψ k ) k [8; 29] . It might be helpful to visualize this bigraded ring as in Fig. 1 , regarding which we offer a few comments: tt-geometry ofétale (Tate) motives • The Milnor K-theory is displayed on the diagonal for visual aid. tt-geometry ofétale (Tate) motives • The darker area always vanishes: for n < 0 or n < m this is for dimension reasons; for n = 0 and m < 0 this follows from a simple computation with higher Chow groups. tt-geometry ofétale (Tate) motives • The lightly shaded area is what we are now interested in: the BeilinsonSoulé vanishing conjecture predicts that the left patch vanishes. It is known in a few cases, for example if F is any of the following: a finite field, a global field (in any characteristic), a function field of a genus 0 curve over a number field. We now introduce an even stronger Hypothesis. We recall the following result on Tate motives with rational coefficients. tt-geometry ofétale (Tate) motives Proposition 6.9 ([33: 4.17] ). Let F be a field satisfying Hypothesis 6.6. Then both morphisms in the following composition are isomorphisms of locally ringed spaces: tt-geometry ofétale (Tate) motives The Beilinson-Soulé part of Hypothesis 6.6 is used to invoke [28] and obtain a bounded t-structure on DTM gm (F, Q). The second part of Hypothesis 6.6 then implies that the tt-spectrum is identified with the "coherent spectrum" of the heart [33: 4.2], which is easily seen to be a singleton. Now we can put the results on torsion and rational coefficients together to arrive at an integral statement (for more general statements see Remark 6.11 and Remark 6.13). tt-geometry ofétale (Tate) motives Theorem 6.10. Let F be a field of exponential characteristic p, and assume that for every p prime, F contains a primitive th root of unity (respectively, 4th root of unity if = 2). Assume also that F satisfies Hypothesis 6.6. Then: tt-geometry ofétale (Tate) motives (1) The central ring R DTM´e Proof. We discussed in Section 4 why p is a unit in the central ring. A more precise statement is [14: 7.1.2] from which the first part of the Theorem follows immediately. tt-geometry ofétale (Tate) motives For the second part, it suffices to show that ρ is a homeomorphism. First we show it is a bijection. For p, the fiber of ρ over is by definition supp(Z/ ). Since Z/ is the image of the unit under γ * , the right adjoint in the adjunction 4) , it suffices to show that the latter has a one point spectrum. This is Proposition 6.9. tt-geometry ofétale (Tate) motives Finally, we need to show that the topologies of the two spaces are the same, i.e. we need to show that the prime m 0 in the generic fiber of ρ is the generic point of Spec(DTM´e t gm (F, Z)). (Here, we use that a bijective spectral map between spectral spaces is a homeomorphism if and only if specializations lift along the map.) Recall that a base for the closed subsets is given by supp(M), where M ∈ DTM´e t gm (F, Z). In other words we need to show that whenever supp(M) contains m 0 then it is the whole space. This is precisely what the second part of Proposition 5.10 says. tt-geometry ofétale (Tate) motives Remark 6.11. Of course, if S ⊂ Z\{0} is a saturated multiplicative subset containing the exponential characteristic of F and such that for each prime S, F contains a primitive th root of unity (respectively, 4th root of unity if = 2) then the same arguments show (still assuming F satisfies Hypothesis 6.6): tt-geometry ofétale (Tate) motives (2) The comparison morphism tt-geometry ofétale (Tate) motives is an isomorphism of locally ringed spaces. Notice in particular that étale cohomology DTM´e tt-geometry ofétale (Tate) motives is conservative (as its kernel is a tt-ideal). Remark 6.13. A statement analogous to Theorem 6.10 (or Remark 6.11) holds for étale Artin-Tate motives if, in addition, Hypothesis 6.6 is satisfied for every finite extension F /F. Indeed, Theorem 6.2 clearly applies for torsion coefficients; and for rational coefficients, [33: 4.17] gives the required result. Filtered Galois representations In this section we recall Positselski's approach to describing Tate motives with torsion coefficients in [35] . The upshot is Proposition 7.7 which tells us that the tt-geometry of Tate motives over certain fields and with torsion coefficients is the same as the tt-geometry of Galois representations with a "unipotent filtration", the latter being seemingly more tractable. Although in this article we will eventually deal with algebraically closed fields only, it does not cause any difficulties to treat the general case here. Filtered Galois representations Convention 7.1. Throughout this section we fix a field F and a prime invertible in F. The absolute Galois group of F is denoted by G F as before. We also assume that F contains a primitive th root of unity ζ ∈ µ (F) which we interpret as a morphism β : Filtered Galois representations To see this identification, notice that by the change of coefficients adjunctions (Eq. For any integer n, define the replete triangulated subcategories DTM gm (F, Z/ ) ≥n (respectively DTM gm (F, Z/ ) <n ) generated by (Z/ )(n ) for n ≥ n (respectively n < n). The vanishing of motivic cohomology H p,q (F, Z/ ) when q < 0 (cf. our remarks regarding Fig. 1 ), implies that the pair Filtered Galois representations defines a t-structure on DTM gm (F, Z/ ). Indeed, the arguments of [28: 1.2] go through word for word. In particular, this gives rise to adjoints W ≥n , W <n to the inclusions of these subcategories, and one deduces that every object M ∈ DTM gm (F, Z/ ) admits a functorial descending filtration, called the weight filtration, Filtered Galois representations such that the associated graded piece gr n M := W ≥n W <n+1 M is a finite direct sum of copies of shifts of Z/ (n) (as H p,q M (F, Z/ ) vanishes when q = 0 and p 0). Define the subcategory F (F, Z/ ) ⊂ DTM gm (F, Z/ ) as the smallest full subcategory containing Z/ (n) for all integers n, and closed under extensions. (The latter condition means that for any triangle in which the outer two terms lie in the subcategory, the middle one does as well.) The Bloch-Kato conjecture (recalled in Section 6) implies the Beilinson-Lichtenbaum conjecture [19; 36] , i.e. the étale realization functor induces identifications: Filtered Galois representations For one, this implies that there are no negative Ext groups between the Tate objects Z/ (n), and from this one deduces that F (F, Z/ ) is in fact an exact subcategory of DTM gm (F, Z/ ) (i.e. the triangles of DTM gm (F, Z/ ) lying in F (F, Z/ ) define an exact structure [see 17]). And secondly, the Beilinson-Lichtenbaum conjecture essentially implies the following result. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories where the latter denotes the category of (discrete) Galois modules over Z/ equipped with a unipotent filtration, i.e. a finite decreasing filtration whose graded pieces are direct sums of copies of the trivial Galois module Z/ . Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories Let us elaborate a bit on the category mod fil-un (G F , Z/ ). The category of Galois modules over Z/ is an abelian tensor category, the tensor product being taken over Z/ . As with any abelian tensor category, the category of its filtered objects mod fil (G F , Z/ ) inherits the structure of an exact tensor category: a conflation is a short sequence a b c with zero composition such that the associated graded sequences 0 → gr n a → gr n b → gr n c → 0 are all short exact in mod(G F , Z/ ). And the tensor product a ⊗ b of two filtered Galois modules a and b has an induced filtration given by Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories The full subcategory mod fil-un (G F , Z/ ) of mod fil (G F , Z/ ) is defined by the triviality condition on the graded pieces. It is clear that if a b c is a conflation and b and c (respectively a and b) satisfy the triviality condition on the graded pieces then so does a (respectively c), and this implies that mod fil-un (G F , Z/ ) is an exact subcategory [35: A.5. (3)]. It is also easy to see that it is closed under tensor products since the associated graded is a tensor functor. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories The étale realization Re sends Z/ (n) to µ (F) ⊗n Z/ , using the primitive th root of unity ζ. Since the heart of D b c (G F , Z/ ) with respect to the standard tstructure is closed under extensions, we see that the image of F (F, Z/ ) under the étale realization is contained in the heart. Applying the realization to the weight filtration Eq. (7.2) of an object M ∈ F (F, Z/ ) therefore yields a discrete Galois module together with a unipotent filtration. This describes the functor in the statement of Proposition 7.3. Note in particular that under this identification, the étale realization corresponds to forgetting the (unipotent) filtration of the discrete Galois module. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories Proof of Proposition 7.3. The fact that this is an equivalence of exact categories is [35: 3.1] . (This obviously relies crucially on the truth of the Beilinson-Lichtenbaum conjecture.) We only want to explain why it is compatible with the tensor structure. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories Let T stand for the category DTM gm (F, Z/ ), and correspondingly T ≥n and T <n for the subcategories considered above. We first note that the tensor product sends T ≥p ×T ≥q into T ≥p+q , as follows immediately from Z/ (m)⊗Z/ (m ) = Z/ (m+m ). Also, note that every object M ∈ T sits in a functorial triangle Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories Now fix integers p, q, and objects M, N ∈ T . The two facts just mentioned imply that the canonical morphism W ≥p M⊗W ≥q N → M⊗N factors through W ≥p+q (M⊗N), and this defines a natural transformation W ≥p ⊗ W ≥q → W ≥p+q • ⊗. There is then an induced morphism Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories using the fact that Re is a tensor functor. By functoriality of this construction, it is obvious that Eq. (7.5) endows Re ,ζ with a lax symmetric unital monoidal structure. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories It remains to check that Eq. (7.5) is in fact invertible when restricted to the category F (F, Z/ ). It is certainly invertible if M = Z/ (m) and N = Z/ (n). Moreover, since the tensor product in DTM gm (F, Z/ ) is exact in both variables, and the étale realization an exact functor, it follows that, in both variables, the set of objects for which Eq. (7.5) is invertible is closed under extensions. Thus the claim. Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories Let us take stock: we have found an exact tensor category F (F, Z/ ) inside the tt-category DTM gm (F, Z/ ) which we understand reasonably well, by Proposition 7.3. The question arises whether DTM gm (F, Z/ ) is simply the (bounded) derived category of F (F, Z/ ). This kind of question is in general very hard to answer, not least because there is no canonical functor in either direction. Typically, this problem can be solved if the triangulated category has a model. In our case we don't know whether DTM gm (F, Z/ ) does, but it is a triangulated subcategory of DM gm (F, Z/ ) which does. This is enough for Positselski to construct an exact functor Proposition 7.3 (cf. [35: 3.1]). The étale realization induces an equivalence of exact tensor categories and he proves that it is an equivalence if (and only if) the graded Galois cohomology algebra H • (G F , Z/ ) is Koszul [35: 9.1]. Unfortunately, it is not clear whether Eq. (7.6) is compatible with the tensor structures, and although it should be possible to construct a tensor equivalence using an extension of the methods employed in [35: Appendix D], we leave it as an open problem for now. Instead, we will establish the following weaker result which is enough for our purposes. Proposition 7.7. Assume that the graded Galois cohomology algebra H • (G F , Z/ ) is Koszul. The étale realization functor together with the exact equivalence of Eq. (7.6) induce a bijection Proposition 7.7. Assume that the graded Galois cohomology algebra H Proof. In order to construct the exact functor of Eq. (7.6), a weaker version of filtered triangulated categories is used in [35] . One can easily phrase the proof to be given here in this language but we prefer to work with derivators which we feel yields a conceptually more satisfying argument. The reason is that in the context of derivators, the bounded derived category of an exact category has the expected universal property [34: 2.17]. We place ourselves in the following abstract situation: ι : A → T is an exact tensor subcategory of a (possibly large) tt-category T which is the base of a stable monoidal derivator T (defined on finite categories). We also assume that Proposition 7.7. Assume that the graded Galois cohomology algebra H The universal property mentioned above yields an exact morphism of derivators F : A → T, where A denotes the derivator with base D b (A). It induces an exact functor F : D b (A) → T which is the identity on A, and is unique up to unique isomorphism for these properties. Now fix an object M ∈ A and consider the two functors Proposition 7.7. Assume that the graded Galois cohomology algebra H They coincide on A, and satisfy the "Toda conditions" [34: 2.17] Proposition 7.7. Assume that the graded Galois cohomology algebra H because M ∈ A and A is closed under tensor products. It follows again from the universal property that the associated functors F 1 , F 2 : A → T are canonically isomorphic. We deduce in particular that the following square commutes on the level of objects: Proposition 7.7. Assume that the graded Galois cohomology algebra H We apply this to A = F (F, Z/ ), and T the derivator associated to the stable monoidal model category modeling DM(F, Z/ ). It follows from the uniqueness statement of [35: A.17] that F : A → T in this case induces an exact equivalence Proposition 7.7. Assume that the graded Galois cohomology algebra H In particular, it induces a bijection of thick subcategories. The proof will be complete once we check that Proposition 7.7. Assume that the graded Galois cohomology algebra H ) are exactly the thick subcategories closed under tensoring with Z/ (n) (i.e. Z/ placed in filtration degree n), n ∈ Z. Proposition 7.7. Assume that the graded Galois cohomology algebra H • tt-ideals in DTM gm (F, Z/ ) are exactly the thick subcategories closed under tensoring with Z/ (n), n ∈ Z. Indeed, Z/ (n) ∈ A; now apply commutativity of the diagram in Eq. (7.8). Proposition 7.7. Assume that the graded Galois cohomology algebra H The two bullet points are an immediate consequence of the fact that in both cases the objects Z/ (n) generate the category as a thick subcategory [see 18: 3.6]. tt-primes In this section we are going to determine the prime ideals in the triangulated category of Tate motives over certain algebraically closed fields. This will use the results in the previous section, as well as the results in [18] where we determined the tt-geometry of filtered modules. As in the étale case (Section 6) we will first treat the case of torsion coefficients; the case of rational coefficients is the same as in the étale case due to the equivalence DTM gm (F, Q) DTM´e t gm (F, Q). Convention 8.1. If not mentioned explicitly otherwise we assume that F is algebraically closed throughout this section. (The only exception is Lemma 8.5.) tt-primes Let be a prime invertible in F, and fix a primitive th root of unity ζ ∈ µ (F) which we interpret as a morphism β : Z/ (0) → Z/ (1) as in Section 7. From Proposition 7.7 and [18: 7.8] we deduce the following result. Proposition 8.2. The tt-category DTM gm (F, Z/ ) has a unique non-trivial tt-ideal given by ker(Re ) = cone(β) . tt-primes Proof. Since F is algebraically closed, G F is the trivial group. Therefore the Galois cohomology algebra H • (G F , Z/ ) is just Z/ , and in particular Koszul. Also, mod fil-un (G F , Z/ ) is canonically identified with the category of filtered (finitedimensional) Z/ -vector spaces, and so its (bounded) derived category is just , and found that it has a unique non-trivial tt-ideal given by ker(π) = cone(β) . 4 The result now follows from Proposition 7.7. tt-primes Fix the invertible object Z/ (1) in DTM gm (F, Z/ ) and define the graded central ring tt-primes is an isomorphism of locally ringed spaces. tt-primes Proof. The first part can be deduced from the Beilinson-Lichtenbaum conjecture, recalled in Section 7. For the second part, it suffices to show that the map is a homeomorphism. The map is a bijection by Proposition 8. Proof. Since Z/ is perfect over Z, the corresponding change of coefficients functor γ * has a right adjoint γ * . By [6: 1.7] , the image of Spc(γ * ) is precisely the support of γ * γ * (Z) = Z/ (0). On the other hand, using Corollary 8.3 we see that the image of Spc(γ * ) is the set of primes in the statement. It follows from Lemma B.6 below that the two primes are distinct since γ * cone(β) ∈ ker(Re )\ ker(γ * ). tt-primes The inclusion ker(γ * ) ⊂ ker(Re ) gives the specialization relation in the statement. (And of course there can be no other specialization relation by continuity of Spc(γ * ).) tt-primes We now have a good understanding of the tt-spectrum of DTM gm (F, R) for coefficients R = Z/ and R
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