Breuil-Kisin-Fargues modules with complex multiplication
Johannes Anschütz
math.NT 11S99, 11G15, 14F30




ja@math.uni-bonn.de



We prove that the category of (rigidified) Breuil-Kisin-Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil-Kisin-Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a “p-adic Serre group”.


Breuil-Kisin-Fargues modules with complex multiplication
    Johannes Anschütz
    December 30, 2023
========================================================





§ INTRODUCTION


In <cit.> L. Fargues introduced an analog, called Breuil-Kisin-Fargues modules, of Breuil-Kisin modules (cf.  <cit.>) over Fontaine's first period ring

    A_inf=W(𝒪_C^♭)

of Witt vectors of the ring of integers 𝒪_C^♭ inside C^♭ where C denotes a non-archimedean, complete and algebraically closed extension of _p and 

    C^♭=_x↦ x^pC
 
its tilt.
To define Breuil-Kisin-Fargues modules let ξ∈ A_inf be a generator of the kernel of Fontaine's map

    θ A_inf→𝒪_C

(cf. <Ref>)
and let

    φ A_inf→ A_inf
 
be the Frobenius of A_inf (induced by the Frobenius of 𝒪_C^♭).
Concretely, a Breuil-Kisin-Fargues modules (M,φ_M) is then a finitely presented A_inf-module M together with an isomorphism

    φ_Mφ^∗(M)[1/φ(ξ)]≅ M[1/φ(ξ)]

such that M[1/p] is finite projective over A_inf[1/p] (cf. <Ref>).
The study of Breuil-Kisin-Fargues modules, which are mixed-characteristic analogs of Drinfeld's shtuka, was taken over in <cit.> and <cit.>. More precisely, in <cit.> to every proper smooth formal scheme 𝔛 over 𝒪_C and every i≥ 0 there is associated a Breuil-Kisin-Fargues module

    H^i_A_inf(𝔛)

interpolating, at least rationally, various cohomology groups attached to 𝔛. Namely, Breuil-Kisin-Fargues modules admit various realizations (cf. <Ref>):
Let (M,φ_M) be a Breuil-Kisin-Fargues module and denote by k the residue field of 𝒪_C. We can associate to M


  * its “étale realization” 

    T:=(M⊗_A_infW(C^♭))^φ_M=1,
 
a finitely generated _p-module,

  * its “crystalline realization” 

    D:=M⊗_A_infW(k),
 
a finitely generated W(k)-module equipped with a φ-semilinear isomorphism φ_D:φ^∗(D[1/p])≅ D[1/p] after inverting p,

  * and its “de Rham realization” 

    V:=M⊗_A_inf,θ𝒪_C,
 a finitely generated 𝒪_C-module. 

If (M,φ)=H^i_A_inf(𝔛) for a proper smooth formal scheme 𝔛/𝒪_C, then the main result of <cit.> shows that these various realizations are, at least after inverting p, given by the étale cohomology H^i_ét(X,_p) of the generic fiber X:=𝔛_C of 𝔛, the crystalline cohomology H^i_(𝔛_0/W(k))[1/p] of the special fiber 𝔛_0⊆𝔛, and the deRham cohomology H^i_(X/C) of X.
Thus we see that Breuil-Kisin-Fargues modules have a “motivic flavour”.
But the general picture surrounding motives is looking for the existence of a tensor functor from smooth projective schemes into some Tannakian category, and the category of Breuil-Kisin-Fargues modules is not Tannakian. In fact, it is even not abelian (cf. <cit.>). To remedy this we follow an idea in <cit.> and introduce rigidifications of Breuil-Kisin-Fargues modules.
Fix a section k→𝒪_C/p of the projection 𝒪_C/p→ k.
If (M,φ_M) is a Breuil-Kisin-Fargues module then a rigidification for (M,φ_M) (cf. <cit.> and <Ref>) is an isomorphism

    α M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_

of φ-modules over B^+_ inducing the identity when base changed to W(k)[1/p]. The first main theorem of this paper is the following (the result is already stated in <cit.>).



The category 
    BKF_rig
 of rigidified Breuil-Kisin-Fargues modules is abelian.   


We remark that the analogous statement for Breuil-Kisin modules is true, but much simpler. The problems for Breuil-Kisin-Fargues modules arise as the ring A_inf is highly non-noetherian. But luckily we can profit from <cit.>, where enough commutative algebra of A_inf-modules is developed.
From <Ref> it is not difficult to deduce that the _p-linear category

    BKF_rig^∘:=_p⊗__pBKF_rig

of rigidified Breuil-Kisin-Fargues modules up to isogeny is Tannakian (cf. <Ref>). The statement is known for Breuil-Kisin modules, and again it is much simpler.
We investigate the Tannakian category BKF_rig^∘ a bit: for example, we prove that it is “connected” (cf. <Ref>) and of homological dimension 1 (cf. <Ref>).

In <Ref> we start to classify rigidified Breuil-Kisin-Fargues modules admitting “complex multiplication”, i.e., Breuil-Kisin-Fargues modules whose Mumford-Tate group in the Tannakian category BKF_rig^∘ is a torus (cf. <Ref> and <Ref>). More concretely, a rigidified Breuil-Kisin-Fargues module (M,φ,α) up to isogeny admits CM if and only if there exists a commutative, semisimple _p-algebra E and an injection E↪End_BKF_rig^∘((M,φ,α)) of _p-algebras such that

    dim__pE=rk_A_inf(M)
 
(cf. <Ref>).
Using the crucial theorem about different descriptions of (finite free) Breuil-Kisin-Fargues modules (cf. <Ref>) due to Fargues/Kedlaya-Liu/Scholze we can then prove our main theorem about the classification of Breuil-Kisin-Fargues modules admitting CM.



For every finite dimensional commutative, semisimple _p-algebra E there exists a (natural) bijection between isomorphism classes of (rigidified) Breuil-Kisin-Fargues modules (up to isogeny) admitting CM by E and functions ΦHom__p(E,C)→.


Moreover, we can write down for given pair (E,ΦHom__p(E,C)→) the corresponding (rigidified) Breuil-Kisin-Fargues module (up to isogeny) explicitly (cf. <Ref>). 
We remark that a similar result has been obtained by Lucia Mocz in the case of Breuil-Kisin modules associated with p-divisible groups (cf. <cit.>).

Finally, let 𝒯⊆BKF^∘_rig be the full Tannakian subcategory spanned by rigidified Breuil-Kisin-Fargues modules up to isogeny admitting CM and let 

    D__p

be the pro-torus over _p with group of characters the coinduced discrete Galois module

    X^∗(D__p)=Coind^Gal(_p/_p)

(cf. <Ref>). We prove the following description of the category 𝒯 of Breuil-Kisin-Fargues modules admitting CM.



The étale realization defines an equivalence

    𝒯≅Rep__p(D__p)
  
of Tannakian categories.


This theorem can be understood as the computation of a “p-adic Serre group”, analogous to the case of rational Hodge structures (cf. <cit.>).

In order to prove <Ref> we develop in <Ref> some language concerning CM-objects and reflex norms in an arbitrary Tannakian category to formalize the known case of rational Hodge structures admitting CM.
 




 §.§ Acknowledgement
 The author wants to thank Lucia Mocz heartily for sharing her notes <cit.> on Breuil-Kisin modules with CM, which eventually led to the question of defining and classifying Breuil-Kisin-Fargues modules with CM answered in this paper. Moreover, the author wants to thank Bhargav Bhatt, Peter Scholze and Sebastian Posur for discussions surrounding this paper. Especial thanks go to Peter Scholze for providing the hint to the remark following <cit.> (i.e., <Ref>) which lead to the construction of the Tannakian category BKF^∘_rig.



§ FORMAL CM-THEORY



In this section we write down a general theory of “CM-objects” in a Tannakian category 𝒯, mainly fixing terminology. Concretely this means that we reformulate the maximal torus quotient of the band of the category 𝒯 internal in 𝒯. We will apply this theory to the case that 𝒯=BKF^∘_rig is the category of rigidified Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>).
The formalism of this section is a straightforward translation of CM-theory for rational Hodge structures or categories of (CM) motives to the case of general Tannakian categories. Therefore we expect it to be known in principle and do not claim any originality.
We advise the reader to have a look at <cit.>.

Let k be a field (later it will assumed to have characteristic 0) and let 𝒯 be a Tannakian category over k (not necessarily assumed to be neutral). For an object X∈𝒯 we denote by ⟨ X⟩^⊗ the full Tannakian subcategory spanned by X.



An object X∈𝒯 is called a CM-object, or to admit CM, if the connected component of the band of the Tannakian category ⟨ X⟩^⊗ is multiplicative, i.e., for every fiber functor

    ω⟨ X⟩^⊗→Bun_S,

where S/k is a scheme, the connected component G^∘ of the group scheme

    G:=Aut^⊗(ω)

is a multiplicative group scheme over S.   


Equivalently, the condition can be required only for the case S=(k^') is the spectrum of a field extension k^'/k, or even only for one fiber functor over some field extension k^'/k.
Moreover, if k^'/k is a finite field extension and 𝒯_k^' the base change of 𝒯 from k to k^' (cf. <cit.>), then X∈𝒯 is a CM-object if and only if k^'⊗_k X∈𝒯_k^' is a CM-object, because a fiber functor ω⟨ X⟩^⊗→Vec_k^'' where k^'' is a field extension of k containing k^' extends to a fiber functor ω^'⟨ k^'⊗_k X⟩^⊗ (cf. <cit.>).  

If k is of characteristic 0, then a connected multiplicative group scheme of finite type is automatically a torus. 

Definition <Ref> is a formalization of the definition of a CM-rational Hodge structure (cf. <cit.>). Namely, let V be a (polarisable) rational Hodge structure and let 𝕊:=Res_ℂ/ℝ𝔾_m be the Deligne torus. Then V is called of CM-type if the “Mumford-Tate group” of V, i.e., the minimal closed subgroup G⊆GL(V) over  containing the image of the morphism h𝕊→GL(V) induced by the Hodge structure V_ℂ=⊕_p,q∈V^p,q on V, is a torus (cf. <cit.>).
This definition agrees with ours as the Mumford-Tate group of V is precisely the automorphism group of the canonical fiber functor

    ω⟨ V⟩^⊗→Vec_.
  





Let T/k be an affine group scheme of finite type which is an extension of a finite discrete group G and multiplicative group T^0, i.e., there exists an exact sequence (of fppf-sheaves)

    1→ T^0→ T→ G→ 1.

Then every V∈_k(T) is a CM-object.   


Let V∈_k(T) and set 𝒯:=⟨ V⟩^⊗ to be the Tannakian subcategory generated by V. Let ω𝒯→Vec_k be the restriction of the canonical fiber functor. Then the canonical morphism

    T→Aut^⊗(ω)
  
is faithfully flat and of finite presentation. This implies that this morphism is open and hence the connected component T^0 of T surjects onto the connected component Aut^⊗(ω)^∘ of Aut^⊗(ω). In particular, the group Aut^⊗(ω)^∘ is multiplicative, i.e., V is a CM-object. 




Let 𝒯 be a Tannakian category over k and let X∈𝒯 be a CM-object. Then every Y∈⟨ X⟩^⊗ is again a CM-object.
Moreover, if X,Y∈𝒯 are CM-objects, then X⊕ Y and X⊗ Y are CM-objects as well.  


For the first statment we may replace 𝒯 by ⟨ X⟩^⊗, and, after enlarging k, assume moreover that 𝒯 is neutral. As X is a CM-object, the category 𝒯≅_k(T) is thus equivalent to the category of representations of an affine group scheme T, which is an extension of a finite discrete group by a multiplicative group as in <Ref>. <Ref> implies that Y∈𝒯=⟨ X⟩^⊗ is again a CM-object. 
Now let X,Y∈𝒯 (with 𝒯 arbitrary) be CM-objects. Then X⊗ Y∈⟨ X⊕ Y⟩^⊗ and thus it suffices to proof that X⊕ Y is again a CM-object by what has already been shown.
But if 

    ω⟨ X⊕ Y⟩^⊗→Bun_S

is a fiber functor, and we let ω_1, resp. ω_2, be the restrictions of ω to ⟨ X⟩^⊗⊆⟨ X⊕ Y⟩^⊗, resp. ⟨ Y⟩^⊗⊆⟨ X⊕ Y⟩^⊗, then the canonical morphism

    Aut^⊗(ω)↪Aut^⊗(ω_1)×Aut^⊗(ω_2)

is a closed immersion. In particular, the connected component of Aut^⊗(⊗) is again multiplicative.  




Let 𝒯 be a Tannakian category over k.
We denote by 𝒯_CM⊆𝒯 the full subcategory of CM-objects of 𝒯.


By <Ref>, the category 𝒯_CM is a Tannakian subcategory. In general, it is not closed under extensions, e.g., for representations of unipotent groups.



Let η𝒯→𝒯^' be an exact tensor functor and let X∈𝒯 be a CM object. Then η(X)∈𝒯^' is a CM object. In particular, η induces an exact tensor functor η𝒯_CM→𝒯^'_CM on the full subcategories of CM-objects.   


Let ω^'⟨η(X)⟩^⊗→Vec_k^' 
be a fiber functor where k^'/k is a field extension. Set ω:=ω^'∘η. The morphism

    η^∗ H:=Aut(ω^')→ G:=Aut(ω)

is then injective. As the connected component of G is multiplicative, the connected component H^∘ of H will therefore be multiplicative as well.


The following direct corollary can be useful to prove that certain Breuil-Kisin-Fargues modules admit CM.



Let T/k be a torus and let ω_k(T)→𝒯 be an exact tensor functor. Then for every object V∈_k(T) the object ω(V) has CM.  


This is a special case of <Ref> as any object in the category Rep_k(T) of representations of T admits CM (cf. <Ref>).  


We now want to give a more explicit definition of CM objects.
From now on assume that k has characteristic 0.
Recall that every object X∈𝒯 in a Tannakian category 𝒯 has a rank

    rk(X)∈End(1_𝒯)≅ k

defined as the trace of the identity on X. In general, the rank is an endomorphism of the unit object 1_𝒯.
But as we assumed that k is of characteristic 0, the rank of X equals the dimension (over k^') of ω(X) for one, or equivalently any, fiber functor ω𝒯→Vec_k^' for k^'/k some field extension.

We call a Tannakian category 𝒯 connected if its band is connected. For a neutral Tannakian category 𝒯≅_k(G), where G/k is an affine group scheme, this is equivalent to saying that G is connected.



Let 𝒯 be a neutral, connected Tannakian category over k (where k is assumed to have characteristic 0). Then an object X∈𝒯 admits CM if and only if there exists a commutative, semisimple k-algebra E of dimension rk(X) and an injection (of k-algebras)

    E↪End_𝒯(X).
   


Assume that X is a CM-object. Then, replacing 𝒯 by ⟨ X⟩^⊗, we may assume that 𝒯≅_k(T) for T/k a torus (as 𝒯 is neutral and connected). In other words, X corresponds to a representation V of the torus T. Decomposing X into a sum of simple objects, we may assume that X is simple. Over an algebraic closure k of k the representation X splits into a direct sum 

    X⊗_kk≅ X_χ_1⊕…⊕ X_χ_n
 
of eigenspaces for distinct characters χ_i T_k→𝔾_m. As X is simple all eigenspaces over k must be one-dimensional. In particular, we see that the endomorphism algebra End_𝒯(X) of X, the formation of which commutes with extensions of k, is a commutative semisimple algebra of dimension the rank of X.
Actually, End_𝒯(X) is a field in this case as X is simple.

Conversely, assume that there exists an injection E↪End_𝒯(X) of a commutative, semisimple k-algebra E such that dim_kE=rk(X). We may pass to a finite extension of k (cf. the remarks after <Ref>) and assume that E≅∏_i=1^nk is isomorphic to copies of k. The idempotents in E define a decomposition X≅⊕_i=1^n X_i with X_i of rank 1 over k. Hence, replacing X by one of the X_i, we may assume that E=k and rk(X)=1 (cf. <Ref>).
But as 𝒯 is neutral, i.e., 𝒯≅_k(G) for some affine group scheme G, a representation ρ G→𝔾_m of rank 1 defines a CM-object X∈𝒯 because the automorphism group scheme of the canonical fiber functor on ⟨ X⟩^⊗ is given by the image ρ(G) of ρ which is multiplicative.  


In general it can happen that for a simple CM-object X in a (non-neutral) Tannakian category in characteristic 0 the endomorphisms

    End_𝒯(X)
  
do not form a field. For example, this happens if 𝒯 is the category of isocrystals over _p. 

We record the following terminology.


If X∈𝒯 is an object in the Tannakian category 𝒯 and ι E↪End_𝒯(X) an injection of a commutative semisimple k-algebra E such that rk(X)=dim_kE, we say that X admits CM by E.   


Now we assume that the Tannakian category 𝒯 is neutral (of characteristic 0) and we fix a fiber functor

    ω_0𝒯→Vec_k.

Moreover, we assume that for some field extension C/k the base extension

    ω_0⊗_k C𝒯→Vec_C

is equipped with a filtration, i.e., we fix a filtered fiber functor

    ω𝒯→FilVec_C

such that ω≅ω_0⊗_k C.
We assume furthermore that C is algebraically closed.
Of course this situation models the case of rational Hodge structures. But it will apply as well to the case of (rigidified) Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>).

In this situation we can define the type of a CM-object.



Let E/k be a commutative, semisimple k-algebra and let 𝒯 be a Tannakian category over k, equipped with fiber functors as above. For a pair (X,ι) with X∈𝒯 an object, necessarily admitting CM, and an injection ι E→End_𝒯(X) such that n:=rk(X)=dim_kE we define the type 

    ΦHom_k(E,C)→

of (X,ι) to be the unique function such that for every i∈

    gr^i(ω(X))≅∏_τ∈Φ^-1(i)C_τ

as a representation of 

    E⊗_kC≅∏_τ∈Hom_k(E,C) C_τ.



If E/k is a commutative semisimple algebra and ΦHom_k(E,C)→ a function, then we call (E,Φ) a CM-type (over k).
Note that

    Hom_k(E,C)≅Hom_k(E,k)

where k⊆ C denotes the algebraic closure of k in C. In particular, the Galois group Gal(k/k) acts naturally on the set

    Hom_k(E,C).




Let (E,Φ) be a CM-type. Then the reflex field E_Φ⊆ C of (E,Φ) is defined to be the fixed field E_Φ⊆k of the stabilizer of ΦHom_k(E,C)→.


If we write E=∏ E_i as a product of fields and thus accordingly Φ=∐Φ_i for functions ΦHom_k(E_i,C)→, then the reflex field E_Φ of (E,Φ) is the composite (in k) of the reflex fields E_Φ_i of the CM-types (E_i,Φ_i).

Let E be a commutative semisimple algebra over k and set T:=Res_E/k𝔾_m to be the Weil restriction of the torus 𝔾_m over E.
Then the group

    X_∗(T):=Hom_k(𝔾_m,k,T_k)

of cocharacters of T is isomorphic (as a Galois module) to the module

    {ΦHom_k(E,C)→}

of types Φ.
Indeed, given a type ΦHom_k(E,C)→ we get the associated cocharacter

    μ_Φ𝔾_m,k→ T_k≅∏_τ∈Hom_k(E,C)𝔾_m,k, t↦ (t^Φ(τ))_τ

of T over k.
By definition the reflex field E_Φ of the type (E,Φ) is the minimal subfield of k over which the cocharacter μ_Φ is defined. In particular, we obtain a cocharacter

    μ_Φ𝔾_m,E_Φ→ T_E_Φ.

In the end, we obtain the reflex norm of (E,Φ) as the composition

    r_ΦRes_E_Φ/k(𝔾_m)Res_E_Φ/k(T_E_Φ) T

where the second morphism denotes the natural norm map.

Let L/k be a finite field extension contained in k. Then we denote by

    L^∗:=Res_L/k(𝔾_m)

the Weil restriction of the multiplicative group 𝔾_m over L to k.
The character group of the torus L^∗ is naturally isomorphic to the Galois module

    [Hom_k(L,k)].

Concretely, let τ L→k be an embedding (over k) and let R/k be a k-algebra. Then the character χ_τ L^∗_k→𝔾_m,k is given on R-points by

    L^∗_k(R)=(R⊗_k L)^×(R⊗_kk)^×R^×

where the right arrow denotes the multiplication R⊗_kk→ R of the k-algebra R.

Let k⊆ L_1⊆ L_2⊆k be a tower of field extensions.
Then there are natural norm maps

    N_L_2/L_1 L_2^∗→ L_1^∗

which on R-valued points for a k-algebra R are given by

    L_2^∗(R)=(R⊗_k L_2)^×   →    (R⊗_k L_1)^×=L_1^∗(R)
    
    x   ↦   det_R⊗_k L_1(x|R⊗_k L_2)

where the determinant is taken of the multiplication by x on the finite free R⊗_k L_1-module R⊗_k L_2.
We define the pro-torus

    D_k:=_L⊆kL^∗

where the transition maps are given by the norms N_L_2/L_1 for L_1⊆ L_2. In particular, the group of characters of D_k is given by

    _L⊆k[Hom_k(L,k)]

where for L_1⊆ L_2 the transition morphism

    [Hom_k(L_1,k)]→[Hom_k(L_2,k)]

send τ L_1→k to the sum

    ∑_τ^' L_2→kτ^'_|L_1=ττ^'.

In particular, we see that the norm morphisms of tori

    N_L_2/L_1 L^∗_2→ L^∗_1

are surjective (as fppf-sheaves).



Let G:=Gal(k/k).
The character group X^∗(D_k) of the pro-torus D_k is canonically isomorphic to the coinduced module

    Coind^G:={f G→ | f  has open stabilizer in  G }
  


This is a general statemenet about discrete G-modules for G a profinite group. Namely, for H⊆ G open,

    Coind^G_H:={f G→ | f  is constant on  H-cosets}

is isomorphic to the free abelian group [G/H] via

    f↦∑_g∈ G/Hf(g)g

and

    Coind^G≅_H⊆ G  openCoind^G_H

by the continuity requirement in the definition of Coind^G.
Moreover, one checks that the transition maps agree under the isomorphism Coind^G_H≅[G/H].


Recall that we have fixed a fiber functor 

    ω_0𝒯→Vec_k

such that its base extension 

    ω:=ω_0⊗_k C𝒯→Vec_C

underlies a filtered fiber functor.



There is a natural tensor functor

    r𝒯_CM→Rep_k(D_k),

which can be explicitly described as follows:
Let X∈𝒯_CM be a CM-object with CM-type (E,Φ) and let E_Φ⊆k be the reflex field of (E,Φ). Then the reflex norm

    r_Φ E_Φ^∗→ E^∗
 
defines an action of the torus E_Φ^∗ on ω_0(X), which defines the action of D_k on ω_0(X) as D_k↠ E_Φ^∗ surjects onto E_Φ^∗.


Let G:=Aut^⊗(ω_0) be the automorphism group of the fiber functor ω_0. Then the graded fiber functor

    gr(ω)𝒯→GrVec_C, X↦⊕_n∈gr^n(ω(X))
 
defines a cocharacter

    μ𝔾_m,k→ G_k

over k (this is spelled out more generally in <cit.>).
Let T be the maximal torus quotient of G, i.e., 

    T=Aut^⊗(ω_0_|𝒯_CM)

is the Tannakian fundamental group of the Tannakian category 𝒯_CM⊆𝒯 of CM-objects in 𝒯. Composition with the canonical morphism G→ T yields the cocharacter

    μ𝔾_m,k→ T_k

of T (which we denote by the same letter).
On character groups this corresponds to a morphism

    μ^∗ X^∗(T)→

of abelian groups. By the universal property of coinduction this defines a Galois-equivariant morphism

    μ^∗ X^∗(T)→Coind^Gal(k/k), χ↦ (g↦χ∘α_g^-1∘μ)

where for g∈Gal(k/k) the morphism α_g T_k→ T_k denotes the action of g on T_k.
In particular, we obtain (using <Ref>) the canonical tensor functor 

    r𝒯_CMRep_k(T)→Rep_k(D_k)

as the morphism induced by the morphism (over k)

    μ^∗ D_k→ T

associated with μ^∗ X^∗(T)→Coind^Gal(k/k)≅ X^∗(D_k). We now identify the functor r with the construction given in the lemma. Let X∈𝒯 be a CM-object with CM-type (E,Φ). Without loss of generality let X be simple. By naturality of the construction in 𝒯 we may further assume that 𝒯=⟨ X⟩^⊗. Then

    E=End_𝒯(X)

(cf. the proof of <Ref>)
and we get

    T=^⊗(ω_0)≅ E^∗

with T acting on X via the action of E^∗.
Indeed, T equals the centralizer C_End_k(ω_0(X))(E^∗) of E^∗ in the endomorphisms of ω_0(X), which itself is E^∗. The containment 

    T⊆ C_End_k(ω_0(X))(E^∗)
 
is clear and equality follows after base change to k where X splits into 1-dimensional representations with distinct characters. Let E_Φ be the reflex field of (E,Φ) (which is given by the stabilizer of μ), and set H:=Gal(k/E_Φ)⊆ G:=Gal(k/k).
In particular, 

    μ𝔾_m,k→ T_k=E^∗_k
 
is already defined over E_Φ.
We have to show that the diagram

    D_k@->>[r][rrd]^μ^∗    E^∗_Φ[r]^-Res_E_Φ/k(μ)   Res_E_Φ/k(T_E_Φ) [d]^Norm
           T=E^∗

commutes. On character groups it gives rise to the diagram

    Coind^G   Coind^G_H()[l]    Coind^G_H(X^∗(T))[l]_μ^∗
           X^∗(T)[llu]^μ^∗[u]

where the arrows without a label are the canonical ones. Here we note that μ^∗ X^∗(T)→ defines an H-equivariant map by the definition of the reflex field (note that the G-module of maps ΨHom_k(E,C)→ is canonically isomorphic to the cocharacter group X_∗(T) of T because T=E^∗). Using the universal property of coinduction it suffices to check that the diagram commutes after composition with the canonical map

    Coind^G()→, f↦ f(1).
 
But then the commutativity follows directly from the definitions.


In the case of rational Hodge structures the above functor yields the classical (connected) Serre group (cf. <cit.>) as a quotient of the pro-torus D_.
In the case of Breuil-Kisin-Fargues modules we will show that the functor r in <Ref> is an equivalence using the following criterion.
We define the full Tannakian subcategory

    𝒯_0⊆𝒯

consisting of all objects X∈𝒯 such that

    gr^i(ω(X))=0

for i≠ 0, where

    ω𝒯→FilVec_C

is our fixed filtered fiber functor.
Moreover, for a finite field extension E/k and an embedding τ E→ C we denote by 

    Φ_τHom_k(E,C)→
 
the map sending τ to 1 and all τ^'≠τ to 0. Clearly, the reflex field of (E,Φ_τ) is τ(E)⊆k⊆ C. 



Assume that the canonical functor Vec_k→𝒯_0 is an equivalence. Then the functor

    r𝒯_CM→Rep_kD_k
  
from <Ref> is fully faithful. Assume moreover that for every finite field extension E/k and every embedding τ E→ C there exists an object X∈𝒯 with CM by (E,Φ_τ). Then the functor r is essentially surjective. If this is the case, then furthermore 𝒯_CM is generated by objects X∈𝒯 with CM by (E,Φ_τ) for E/k a finite field extension and τ E→ C an embedding.


We can assume 𝒯_CM=𝒯.
Define

    T:=Aut^⊗(ω_0)

as the torus representing the tensor automorphisms of the fixed fiber functor

    ω_0𝒯→Vec_k.

Then 𝒯≅Rep_k(T) and the functor r in <Ref> is given by the morphism

    μ^∗ D_k→ T

(in the notation of <Ref>) corresponding to the morphism

    μ^∗ X^∗(T)→ X^∗(D_k)≅Coind^G()

on character groups. Let

    μ𝔾_m,k→ T_k

be the cocharacter induced by the (graded fiber functor associated with the) filtered fiber functor ω and let

    μ^∗ X^∗(T)→

be the induced morphism on characters.
The assumption that the subcategory 𝒯_0 is equivalent to the category Vec_k of k-vector spaces implies that the kernel Ker(μ^∗) contains no non-trivial orbit under the Galois group 

    G:=Gal(k/k).
 
Indeed, if the orbit under some non-trivial χ∈ X^∗(T) under G lies in ker(μ^∗), then the sum

    ∑_g∈ G/Stab_G(χ)gχ

defines a representation X∈Rep_k(T)≅𝒯 of T over k such that gr^i(ω(X))=0 for i≠ 0. In other words, X∈𝒯_0. But this is a contradiction as T acts by assumption trivially on any object in 𝒯_0. Thus the kernel Ker(μ^∗) does not contain a non-trivial G-orbit.
The kernel Ker(μ^∗) contains the kernel Ker(μ^∗). Moreover, Ker(μ^∗) is stable under G and hence trivial as it does not contain a non-trivial G-orbit, as well. Hence, the morphism

    X^∗(T)→Coind^G()

is injective and therefore the morphism D_k→ T surjective (as a morphism of fppf-sheaves). This implies that the functor r𝒯→Rep_k(D_k) is fully faithful. The character group 

    Coind^G()≅_L/k[Hom_k(L,C)]
 
of D_k is generated (as a Galois module) by embeddings τ E→ C for finite field extensions E/k (more precisely, by the canonical embedding L⊆ C for L:=τ(E)). Let X∈𝒯_CM be an object with CM by (E,Φ_τ). Then the reflex field E_Φ_τ of the CM-type (E,Φ_τ) is given by

    E_Φ_τ:=τ(E)⊆k⊆ C.

Moreover, the reflex norm

    r_Φ_τ E_Φ_τ^∗E^∗

is induced by the inverse of τ E≅ E_Φ_τ. Indeed, let H=Gal(k/E_Φ_τ)⊆ G be the stabilizer of Φ_τ.
Then the reflex norm E_Φ_τ^∗→ E^∗ corresponds to the G-invariant morphism

    [Hom_k(E,C)]→Coind^G_H([Hom_k(E,C)])Coind^G_H≅[Hom_k(E_Φ_τ,C)]

on character lattices and τ∈Hom_k(E,C) is sent to the canonical inclusion 

    E_Φ_τ⊆ C.

In particular, we can conclude that X must be simple. Namely, its image 

    r(X)∈Rep_k(D_k)

is given by the k-vector space E with D_k through its quotient E_Φ_τ^∗ acting via τ^-1 E_Φ_τ^∗→ E^∗.
In particular, r(X) and hence X are simple.
As in the proof of <Ref> we see that T↠ E^∗ surjects onto the torus E^∗ and thus the functor

    Rep_k(E^∗)→𝒯

is fully faithful. 
As the above morphism

    [Hom_k(E,C)]≅[Hom_k(E_Φ_τ,C)]
  
is an isomorphism we see that 

    [Hom_k(E_Φ_τ,C)]⊆Coind^G

lies in the image of X^∗(T). As E/k was arbitrary, we can conclude that

    X^∗(T)≅Coind^G

which implies

    𝒯≅Rep_k(D_k).

For the last statement it suffices to note, as was mentioned above, that the character group

    Coind^G()≅_k/L/k[Hom_k(L,C)]

is generated by embeddings τ L→ C for L/k finite.




§ BREUIL-KISIN-FARGUES MODULES UP TO ISOGENY



We try to follow closely the notations in <cit.> and <cit.>.
Fix a prime p.
Let C/_p be a complete, non-archimedean, algebraically closed extension of _p, e.g., C=ℂ_p=_p the completion of an algebraic closure of _p.
Let 𝒪_C⊆ C be the ring of integers in C and let

    C^♭:=_x↦ x^pC
  
be the tilt of C with its ring of integers

    𝒪_C^♭=𝒪_C^♭=_x↦ x^p𝒪_C≅_x↦ x^p𝒪_C/p.

We denote by

    (-)^# C^♭→ C, (x_0,x_1,…)↦ x_0

resp.
    (-)^#𝒪_C^♭→𝒪_C, (x_0,x_1,…)↦ x_0

the canonical projections.
Let 

    ϵ:=(1,ζ_p,ζ_p^2,…)∈𝒪_C^♭
 
be a fixed system of p-power roots of unity in C^♭ (the results of this paper will be independent of the choice of ϵ).
Define

    A_inf:=W(𝒪_C^♭)

as the ring of Witt vectors for the (perfect) ring 𝒪_C^♭.
We denote by

    [-]𝒪_C^♭→ W(𝒪_C^♭)

the canonical Teichmüller lift.
When dealing with A_inf as a topological ring we always equip it with the (p,[ϖ])-adic topology for a pseudo-uniformizer ϖ∈ C^♭ (this topology is independent of the choice of ϖ). 
Let

    φ A_inf→ A_inf, ∑_i=0^∞ [x_i]p^i↦∑_i=0^∞ [x_i^p]p^i

be the Frobenius on A_inf.
We define the elements

    μ:=[ϵ]-1
    ξ:=[ϵ]-1/[ϵ^1/p]-1=∑_i=0^p-1[ϵ^i/p-1]
    ξ̃:=φ(ξ)=[ϵ^p]-1/[ϵ]-1

of A_inf.
Let

    θ A_inf→𝒪_C, ∑_i=0^∞ [x_i]p^i↦∑_i=0^∞ (x_i)^#p^i

be Fontaine's map θ and set

    θ̃:=θ∘φ^-1.

Then ξ∈ A_inf is a generator of (θ)
and ξ̃∈ A_inf a generator of (θ̃).
We define as usual

    A_

as the universal PD-thickening of θ and B^+_:=A_[1/p].
Finally, Fontaine's period ring B^+_ is defined to be the ξ-adic completion of A_inf[1/p]. In particular, the map θ induces a (surjective) morphism

    θ B^+_→ C.

In fact, the ring B^+_ is a complete discrete valuation ring with uniformiser ξ and residue field C. Its fraction field B_:=B^+_[1/ξ] is also called the field of p-adic periods.

Having fixed this notation we can now define the notion of a Breuil-Kisin-Fargues module which is a “perfectoid” analog of a Breuil-Kisin module.



 A Breuil-Kisin-Fargues module is a finitely presented A_inf-module M with an isomorphism

    φ_M (φ^∗ M)[1/ξ̃]=M⊗_A_inf,φA_inf[1/ξ̃] M[1/ξ̃]
 
such that M[1/p] is a finite projective A_inf[1/ξ̃]-module.


By <cit.> a finite projective A_inf[1/p]-module is automatically free, hence in <Ref> the condition “finite projective” can be replaced by “finite free".
We denote by

    BKF

the category of Breuil-Kisin-Fargues modules (cf. <cit.>). It is (naturally) an exact tensor category. However, it is not abelian, only pseudo-abelian.



Let (M,φ_M) be a Breuil-Kisin-Fargues module and let e M→ M be an idempotent endomorphism of (M,φ_M), i.e., e commutes with φ_M. Then ker(e) and coker(e) with their induced Frobenii are Breuil-Kisin-Fargues modules. In particular, the category BKF of Breuil-Kisin-Fargues modules is pseudo-abelian.   


Clearly, the kernel and cokernel of e admit Frobenii. As direct summands of finitely presented (resp. finite projective) modules are again finitely presented(resp. finite projective) ker(e) and coker(e) satisfy all conditions in <Ref>, i.e., they are Breuil-Kisin-Fargues modules.


We give an easy example of a Breuil-Kisin-Fargues modules.


For d∈ we set A_inf{d}:=μ^-dA_inf⊗__p_p(1) with Frobenius 

    φ_A_inf{d}φ^∗(A_inf{d})=φ(μ)^-dA_inf⊗__p_p(1)→ A_inf{d}, a↦ξ̃^d a






Let (M,φ_M) be a Breuil-Kisin-Fargues module such that M is finite projective (equivalently, free) over A_inf. Then the dual (M^∨,φ_M^∨) of (M,φ_M) is defined by

    M^∨:=Hom_A_inf(M,A_inf)

with Frobenius

    φ_M^∨ (φ^∗ M^∨)[1/ξ̃]=(φ^∗ M)[1/ξ̃]^∨(M[1/ξ̃])^∨=M^∨[1/ξ̃].

where we used the notation (-)^∨=Hom_A_inf[1/ξ̃](-,A_inf[1/ξ̃]) again for duals of finite projective A_inf[1/ξ̃]-modules.  


We recall the following lemma about general A_inf-modules.



Let M be a finitely presented A_inf-module such that M[1/p] is finite projective (equivalently, free) over A_inf. Then there is a functorial (in M) exact sequence

    0→ M_tor→ M→ M_free→M→ 0

satisfying:


  i) M_tor is finitely presented and perfect as an A_inf-module, and killed by p^n for some n≫ 0.

  ii) M_free is a finite free A_inf-module.

  iii) M is finitely presented and perfect as an A_inf-module, and is supported at the closed point of (A_inf). 



See <cit.>. 


In particular, a finitely presented A_inf-module M such that M[1/p] is finite free over A_inf[1/p] is perfect as an A_inf-module, i.e., admits a finite projective resolution (cf. <cit.>). 

Moreover, if the A_inf-module M in <Ref> is a Breuil-Kisin-Fargues module, i.e., equipped with a Frobenius 

    φ_Mφ^∗(M)[1/ξ̃]≅ M[1/ξ̃],
 
then the modules M_rm, M_free and M carry a natural Frobenius as well and the exact sequence in <Ref> is an exact sequence of Breuil-Kisin-Fargues modules (note that M_tor[1/p]=M[1/p]=0 is free). In fact, the existence of this Frobenius is clear for M_tor. But M_free=H^0((A_inf),M/M_tor) (cf. the proof of <cit.>) which yields the Frobenius on M_free (and thus M as well). 

We want to recall some equivalent descriptions of Breuil-Kisin-Fargues modules whose underlying A_inf-module is finite free (cf. <cit.>). For this we need to recall that associated to the fixed complete and algebraically closed non-archimedean field C/_p there is associated a scheme X_FF over _p, “the Fargues-Fontaine curve”, together with a distinguished point ∞∈ X_FF whose completed stalk 𝒪_X_FF,∞≅ B^+_ is isomorphic to Fontaine's period ring B^+_ associated with C (cf. <cit.>). We also recall that the adic Fargues-Fontaine curve X_FF^=Y/φ^ is uniformized by an adic space Y admitting an action of φ (cf. <cit.>).
The space Y is defined to be 

    Y:=Spa(A_inf)∖ V(p[ϖ])

where ϖ∈𝒪_C^♭ is a pseudo-uniformizer (cf. <cit.> for a proof that this is actually an honest adic space, i.e., the structure presheaf is a sheaf).
Additionally to Y the adic space Spa(A_inf)∖ V(p,[ϖ]) contains two points, namely


  * x_ given as the image of the p-adic valuation on W(k) along the canonical projection A_inf→ W(k) (this point is denoted x_L in <cit.>).

  * x_étale given by the image of the valuation on 𝒪_C^♭ along the morphism A_inf→ O_C^♭=A_inf/p (this point is denoted x_C^♭ in <cit.>). 


We set

    B:=H^0(Y,𝒪_Y)

and 

    B^+:=H^0((A_inf)∖V(p), 𝒪).

There exists a natural φ-equivariant injection B^+→ B^+_ making B^+_ a B^+-algebra (cf. <cit.>).



The category Bun_X_FF of vector bundles on the Fargues-Fontaine curve is naturally equivalent to the category of φ-modules over B^+_, i.e., to finite projective B^+_-modules M together with an isomorphism φ_Mφ^∗ M≅ M.  


This is proven in <cit.>. The natural functor inducing the equivalence is given by pulling back a vector bundle ℱ∈Bun_X_FF≅_X^_FF to Y, extending it along the point x_ and tensoring the global sections of this extension (which form a φ-module over B^+) over B^+ with B^+_.



 
The following categories are equivalent:


  i) Finite free Breuil-Kisin-Fargues modules (M,φ_M).

  ii) Pairs (T,Ξ), where T is a finite free _p-module, and Ξ⊆ T⊗__pB_ is a B^+_-lattice.

  iii) Quadruples (ℱ,ℱ^',β,T), where ℱ and ℱ^' are vector bundles on the Fargues-Fontaine curve X_FF, and βℱ_|X_FF∖∞ℱ^'_|X_FF∖∞ is an isomorphism, ℱ is trivial, and T⊆ H^0(X_FF,ℱ) is a _p-lattice.  



The proof is given in <cit.>. We shortly give descriptions of the various functors involved.
Let (M,φ_M) be a finite free Breuil-Kisin-Fargues module. Then the φ_M-invariants

    T:=(M⊗_A_infW(C^♭))^φ_M=1

form a finite free _p-module (by Artin-Schreier theory, cf. <cit.>). Note that ξ̃ is invertible in W(C^♭), hence the base extension M⊗_A_infW(C^♭) actually carries a Frobenius. Moreover, multiplication defines an isomorphism

    T⊗__pW(C^♭)≅ M⊗_A_infW(C^♭)

under which T⊗__pA_inf[1/μ]⊆ T⊗__pW(C^♭) is mapped to M⊗_A_infA_inf[1/μ] (cf. <cit.>).
Setting 

    Ξ:=M⊗_A_infB^+_⊆ (M⊗_A_infA_inf[1/μ])⊗_A_inf[1/μ]B_≅ T⊗__pB_
 defines the pair (T,Ξ) in ii) associated with (M,φ_M). If Ξ⊆ T⊗__pB_ is a B^+_-lattice, where T is a finite free _p-module, then (in order to pass to iii)) one can use this lattice to modify the trivial bundle ℱ:=T⊗__p𝒪_X_FF at the distinguished point ∞∈ x_FF on the Fargues-Fontaine curve to obtain a bundle ℱ^' with an isomorphism ℱℱ^' away from ∞ (recall that 𝒪_X_FF,∞≅ B^+_).
Clearly, T⊆ H^0(X_FF,𝒪_X_FF) defines a _p-lattice as H^0(X_FF,𝒪_X_FF)≅ T⊗__p_p. Moreover, this construction can be reversed giving the equivalence between the categories in ii) and iii). Thus in particular, the hard part of the theorem is the construction of a functor from the category in ii) (or iii)) to Breuil-Kisin-Fargues modules. We shortly describe how this is done. Let (ℱ,ℱ^', α,T) be given as in iii). Pulling back ℱ and ℱ^' to Y defines two φ-equivariant vector bundles ℰ,ℰ^' on Y. By triviality of ℱ the vector bundle ℰ extends (non-canonically) to the point x_étale. This extension is not unique, but depends on the choice of a _p-lattice in H^0(X,ℱ)=H^0(Y,ℰ)^φ=1. In particular, the data of T yiels a canonical extension of ℰ to x_étale. Moreover, every φ-equivariant vector bundle on Y, thus in particular ℰ^', extends uniquely to the point x_.
Using the modification ℱℱ^' away from ∞ the bundles ℰ and ℰ^' are (φ-equivariantly) isomorphic when restricted to a small annulus omitting the points φ^n(∞),n∈. In particular, one obtains a vector bundle ℰ^'' on (A_inf)∖ V(p,[ϖ]) which restricts to the extension of ℰ^' near x_ and to the extension of ℰ near x_étale. Taking global sections of ℰ^'' defines the finite free A_inf-module M which is then moreover equipped with a Frobenius φ_M away from V(ξ̃), i.e., a finite free Breuil-Kisin-Fargues module.
In particular, we see that the φ-module M⊗_A_infB^+_ over B^+_ is canonically isomorphic to the φ-module over B^+_ associated with ℱ^' as in <Ref>.   


For example, the Breuil-Kisin-Fargues modules A_inf{d} is sent to the pair

    (_p(d),Ξ:=ξ^-d(_p(d)⊗__pB^+_)⊆_p(d)⊗__pB_)

resp., for d≥ 0, to the modification

    0→_p(d)⊗__p𝒪_X_FF𝒪_X_FF(d)→ t^-d𝒪_X_FF,∞/𝒪_X_FF,∞→ 0

induced by the d-th power t^d of Fontaine's t=log([ϵ])∈ H^0(X_FF,𝒪_X_FF(1))=(B^+_)^φ=p.

We want to stress that the categories in <Ref> are not equivalent as exact categories, i.e., the equivalences (or their inverses) are not exact. In fact, in the construction of the finite free Breuil-Kisin-Fargues module one has to extend (by taking global sections) a vector bundle on the punctured (A_inf) with the closed point removed to the whole of (A_inf) and this is not an exact operation.



We define the isogeny category of Breuil-Kisin-Fargues modules BKF^∘ as 

    BKF^∘:=_p⊗__pBKF,

i.e., the category BKF^∘ has the same objects as the category BKF, but for M,N∈BKF^∘ the space of homomorphisms is given by

    Hom_BKF^∘(M,N):=_p⊗__pHom_BKF(M,N).
    


The category BKF^∘ is still not abelian (for the same reason as BKF, cf. <cit.>), but rigid as we now show.



The isogeny category BKF^∘ of Breuil-Kisin-Fargues modules is a rigid, exact tensor category.  


It suffices to prove that every object admits a dual because the tensor product and the exact structure are inherited from the category BKF. Let (M,φ_M)∈BKF be a Breuil-Kisin-Kisin-Fargues module. We claim that, in the isogeny category, (M,φ_M) is isomorphic to a Breuil-Kisin-Fargues module (N,φ_N) such that N is finite free over A_inf. Let M_tor⊆ M be the torsion submodule. Then φ^∗(M_tor) resp. φ^∗(M_tor)[1/ξ̃] is the torsion submodule of φ^∗(M) resp. φ^∗(M)[1/ξ̃]. Therefore it is stable by φ_M. By lemma <Ref> the torsion submodule M_tor is actually killed by p^n for n≫ 0, and thus zero in the isogeny category. Hence, we may assume that M is actually torsion-free. Set N:=M_free as in <Ref>, which is finite free over A_inf. Then φ^∗(N)[1/ξ̃]=φ^∗(M)[1/ξ̃] and N[1/ξ̃]=M[1/ξ̃] because the cokernel of M→ N is killed by ξresp.  ξ̃.
In particular, the Frobenius φ_M of M defines a Breuil-Kisin-Fargues module (N,φ_M) and we are finally allowed to replace M by N, as (M,φ_M) and (N,φ_N) are isomorphic in the isogeny category. This finishes the proof as finite free Breuil-Kisin-Fargues modules clearly admit duals.  


We remark that the proof of <Ref> actually shows that for a Breuil-Kisin-Fargues module (M,φ_M) the exact sequence

    0→ M_tor→ M→ M_free→M→ 0

of <Ref> can be enhanced to an exact sequence of Breuil-Kisin-Fargues modules, i.e., there are compatible Frobenii on the modules involved.

In terms of pairs of a _p-lattice and a B^+_-lattice the category BKF^∘ of Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of pairs (V,Ξ) where V is a finite dimensional _p-vector space and Ξ⊆ V⊗__pB_ a B^+_-lattice. Again, we can see that this category is not abelian: The canonical morphism

    (_p,ξ B^+_)→ (_p,B^+_)

does not admit a cokernel. However, we see that it admits kernels. But we caution the reader that it is not clear how to describe the kernel of a morphism f (M,φ_M)→ (N,φ_N) of Breuil-Kisin-Fargues modules up to isogeny, because it is not clear whether the kernel K:=Ker(MN) is finitely presented over A_inf or satisfies the property that K[1/p] is finite projective over A_inf[1/p] (cf. the proof of <Ref>).

We are seeking for a Tannakian, and thus in particular abelian, category of Breuil-Kisin-Fargues-modules up to isogeny. This is possible after adding a rigidification after base change to B^+_=A_[1/p] (cf. <cit.>).
Let us recall <cit.>.



Let (M,φ_M) be a Breuil-Kisin-Fargues module. Then M=M⊗_A_infW(k) is a finitely generated W(k)-module equipped with a Frobenius automorphism after inverting p. Fix a section k→𝒪_C/p, which induces a section W(k)→ A_inf. Then there is a (noncanonical) φ-equivariant isomorphism

    M⊗_A_infB^+_≅M⊗_W(k)B^+_
 
reducing to the identity over W(k)[1/p].


See <cit.> which refers to <cit.>.  


We remark that ξ̃ is invertible in B^+_ (but ξ is not), hence the Frobenius φ_Mφ^∗M[1/ξ̃] M[1/ξ̃] defines an isomorphism

    φ_M⊗φ_B^+_φ^∗(M⊗_A_inf B^+_) M⊗_A_infB^+_.


We fix a section k→𝒪_C/p of the projection 𝒪_C/p→ k. This yields a section W(k)→ A_inf of the projection A_inf→ W(k).


  
A rigidified Breuil-Kisin-Fargues module is a Breuil-Kisin-Fargues module (M,φ_M) together with a φ-equivariant isomorphism

    α M⊗_A_infB^+_ (M⊗_A_infW(k))⊗_W(k)B^+_

such that α reduces to the identity over W(k)[1/p], i.e., 

    α⊗_B^+_W(k)[1/p]=_M⊗_A_infW(k)[1/p].


We call an isomorphism α as in <Ref> a rigidification of (M,φ_M).
We denote by

    BKF_rig

the category of rigidified Breuil-Kisin-Fargues modules, i.e., its objects are rigidified Breuil-Kisin-Fargues modules and the morphisms are morphisms of Breuil-Kisin-Fargues modules respecting the given rigidifications.
We illustrate the effect of imposing a rigidification in the case of Breuil-Kisin-Fargues modules of rank 1.



Each finite free Breuil-Kisin-Fargues modules of rank 1 is isomorphic to
 the Breuil-Kisin-Fargues module A_inf{d} for some d∈. For d,d^'∈ we have

    Hom_BKF(A_inf{d},A_inf{d^'})≅_p     if  d≤ d^'
    
    0     otherwise

while

    Hom_BKF_rig(A_inf{d},A_inf{d^'})≅_p     if  d= d^'
    
    0     otherwise



The first statement and the computation of Hom_BKF(A_inf{d},A_inf{d^'}) follows from <Ref> (and the example following it). To prove

    Hom_BKF_rig(A_inf{d},A_inf{d^'})=0
 
if d≠ d^' it suffices to show that the isocrystal (A_inf{d}⊗_A_infW(k)[1/p],φ_A_inf{d}⊗Id) has slope d because there are no morphisms between isocrystals of different slopes.
But the image of 
    ξ̃=[ϵ^p]-1/[ϵ]-1=1+[ϵ]+… + [ϵ^p-1]
 in W(k) is p because k does not contain non-trivial p-power roots of unity.
Hence, the isocrystal (A_inf{d}⊗_A_infW(k)[1/p],φ_A_inf{d}⊗Id) is isomorphic to (W(k)[1/p],p^dφ) which is of slope d. 


The category BKF_rig of rigidified Breuil-Kisin-Fargues modules is again an exact tensor category with the exact structure and tensor product inherited from BKF (taking the tensor product of the rigidifications), but it is moreover abelian (cf. <Ref>) contrary to the case of the category BKF of non-rigidified Breuil-Kisin-Fargues modules.

To prove this we recall the following criterion for an A_inf-module M to satisfy that M[1/p] is finite projective over A_inf[1/p].
Recall the element μ=[ϵ]-1∈ A_inf (cf. the notation in the beginning of <Ref>).



Let M be a finitely presented A_inf-module. Assume


  i) M[1/pμ] is finite projective over A_inf[1/pμ].

  ii) M⊗_A_infB^+_ is finite projective over B^+_. 

Then M[1/p] is finite free over A_inf[1/p].


See <cit.>.  


We record the following corollary in the case of Breuil-Kisin-Fargues modules.


 
Let f (M,φ_M)→ (N,φ_N) be a morphism of Breuil-Kisin-Fargues modules and let Q:=Coker(M N) be the cokernel (of A_inf-modules). Then
Q[1/pμ] is a finite free A_inf[1/pμ]-module.


By <cit.> there is a canonical isomorphism

    M⊗_A_infA_inf[1/μ]≅ T_M⊗__pA_inf[1/μ]

with T_M:=(M⊗_A_infW(C^♭))^φ_M=1
and similarly for N,

    N⊗_A_infA_inf[1/μ]≅ T_N⊗__pA_inf[1/μ],

with T_N:=(N⊗_A_infW(C^♭))^φ_N=1.
In particular, we get

    Q⊗_A_infA_inf[1/pμ]≅ Q^'⊗__pA_inf[1/pμ]

where Q^' is the cokernel of the morphism T_M⊗__p_p→ T_N⊗__p_p induced by f.
Thus we see that

    Q⊗_A_infA_inf[1/pμ]

is finite free.


We remark that in general Q (with the induced Frobenius from (N,φ_N)) will not be a Breuil-Kisin-Fargues modules, because Q[1/p] need not be a finite projective (equivalently, free) A_inf[1/p]-module. The purpose of introducing the rigidifications is exactly to ensure this freeness (cf. the remark after <cit.>).  


  
The category BKF_rig of rigidified Breuil-Kisin-Fargues modules is an abelian tensor category.


It suffices to prove that BKF_rig is abelian.
Let (M,φ_M,α_M) and (N,φ_N,α_N) be two rigidified Breuil-Kisin-Fargues modules with Frobenii φ_M resp. φ_N (after inverting ξ̃) and rigidifications α_M and α_N. Let f (M,φ_M,α_M)→ (N,φ_N,α_N) be a morphism of rigidified Breuil-Kisin-Fargues modules. Consider the exact sequence

    0→ K→ MN→ Q→ 0

of A_inf-modules. We will show that the kernel K and the cokernel Q are finitely presented over A_inf and admit Frobenii φ_K, φ_Q (after inverting ξ̃) and rigidifications α_K, α_Q induced from φ_M, φ_N resp. α_M,α_N, such that (K,φ_K,α_K) and (Q,φ_Q,α_Q) are rigidified Breuil-Kisin-Fargues modules. This will then imply that the category BKF_rig is abelian. In particular, we have to prove that K[1/p] and Q[1/p] are finite free.
As f commutes with φ_M resp. φ_N it is clear that K and Q are canonically equipped with Frobenii φ_K and φ_Q.
Clearly, the module Q is finitely presented. By <Ref> the module Q[1/pμ] is finite projective over A_inf[1/pμ]. Moreover, as f respects the rigidifications

    α_M M⊗_A_infB^+_≅M⊗_W(k)B^+_

resp.
    α_N N⊗_A_infB^+_≅N⊗_W(k)B^+_,

where M:=M⊗_A_infW(k) resp. N:=N⊗_A_infW(k),
 we see that Q⊗_A_infB^+_ is isomorphic to the cokernel of

    M⊗_W(k)B^+_→N⊗_W(k)B^+_,

which is free because the cokernel of M[1/p]→N[1/p] is free as W(k)[1/p] is a field. By <Ref> we can conclude that Q[1/p] is finite free over A_inf[1/p]. As moreover, M[1/p] and N[1/p] are finite free over A_inf[1/p], we can conclude that K[1/p] is finite projective (and hence finite free).
Let C be the two-term complex (concentrated in degree 0 and 1) 

    …→ M N→…

which is a perfect complex of A_inf-modules satisfying that its cohomology if finite free over A_inf[1/p]. By <cit.> we can conclude that H^0(C)=K is finitely presented.
In other words, we have proven that (K,φ_K) and (Q,φ_Q) are Breuil-Kisin-Fargues modules and are left with showing that they admit canonical rigidifications. By right exactness of tensor products this is clear for Q. But as K,M,N,Q are free after inverting p we get a commutative diagram

    0[r]    K⊗_A_infB^+_[r][d]^α_K    M⊗_A_infB^+_[r][d]^α_M    N⊗_A_infB^+_[d]^α_N
    
    0[r]    K⊗_W(k)B^+_[r]   M⊗_W(k)B^+_[r]   N⊗_W(k)B^+_

whose rows are exact. In particular, we also get a canonical rigidifications α_K on K. This finishes the proof of the theorem.


We are now ready to define a main player of this paper.



We define the isogeny category of rigidified Breuil-Kisin-Fargues modules to be the category

    BKF_rig^∘:=_p⊗__pBKF_rig.
  


In other words, the category BKF_rig^∘ is the Serre quotient of the abelian category BKF_rig by the full subcategory of rigidified Breuil-Kisin-Fargues modules (M,φ_M,α_M) such the underlying A_inf-module M is annihilated by p^n, n≫ 0.



The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of triples (N,φ_N,α_N) where


  * N is a finite free A_inf[1/p]-module

  * φ_Nφ_A_inf[1/p]^∗(N)[1/ξ̃] N[1/ξ̃] is an isomorphism and

  * α_N N⊗_A_inf[1/p]B^+_≅N⊗_W(k)[1/p]B^+_ is a rigidification reducing to the identity over W(k)[1/p] with N:=N⊗_A_inf[1/p]W(k)[1/p]

such that there exists a finitely presented A_inf-submodule N^'⊆ N satisfying N^'[1/p]=N which is stable under φ_N.


The natural functor sending an object (M,φ_M,α_M)∈BKF_rig^∘ to its base extension (M⊗_A_infA_inf[1/p],φ_M⊗φ_A_inf[1/p],α_M) to A_inf[1/p] is essential surjective by the last condition on the existence of N^'. It is moreover fully faithful as inverting p commutes with Hom for finitely presented modules.    


The condition on the existence of a φ-stable A_inf-lattice N^'⊆ N ensures that for a triple (N,φ_N,α_N) as in <Ref> the “etale realization” (cf. <Ref>)

    (N⊗_A_infW(C^♭))^φ_N=1
  
is a _p-vector space of dimension the rank of N. Otherwise, the isocrystal N⊗_A_infW(C^♭) (over C^♭) need not be isoclinic of slope 0.

We can now record the main theorem of this section.



The category BKF_rig^∘ of rigidified Breuil-Kisin-Fargues modules up to isogeny is Tannakian.  


Theorem <Ref> implies that also the isogeny category BKF_rig^∘ of rigidified Breuil-Kisin-Fargues modules is abelian, namely it identifies with the Serre quotient of the category BKF_rig by the full subcategory of rigidified Breuil-Kisin-Fargues modules which are p-power torsion.
By (the proof of) <Ref> each object (M,φ_M)∈BKF^∘ is isomorphic (in the isogeny category) to some (N,φ_N) with N a finite free A_inf-module. If

    α_M M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_

is a rigidification for (M,φ_M), then α_M also defines a rigidification of (N,φ_N) because M⊗_A_infB^+_≅ N⊗_A_infB^+_ resp. (M⊗_A_infW(k))⊗_W(k)B^+_≅ (N⊗_A_infW(k))⊗_W(k)B^+_.
In particular, we see that every object in BKF_rig^∘ admits a dual and therefore the category of BKF_rig^∘ is rigid. To prove that BKF_rig^∘ is Tannakian we can either apply Deligne's criterion that Λ^n M vanishes for every M∈BKF^∘_rig and n≫ 0 (cf. <cit.> or use that there are various concrete fiber functors <Ref>. 



We continue by giving an alternative description of rigidified Breuil-Kisin-Fargues modules in terms of modifications of vector bundles on the Fargues-Fontaine curve.

We recall that the Fargues-Fontaine curve admits an Harder-Narasimhan formalism, namely, for every vector bundle ℱ on X_FF there exists a canonical (decreasing) filtration indexed by λ∈

    HN^λ(ℱ)⊆ℱ

such that the associated graded pieces

    gr^λ(HN(ℱ))

are semistable of slope λ.


  
The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules is equivalent to the category of quadruples (ℱ,ℱ^',β,α) where ℱ, ℱ^' are vector bundles on the Fargues-Fontaine curve X_FF, ℱ is trivial, βℱ_|X_FF∖{∞}≅ℱ^'_|X_FF∖{∞} is an isomorphism, and αℱ^'≅⊕_λ∈gr^λ(HN(ℱ^')) is an isomorphism inducing the identity on all graded pieces of the Harder-Narasimhan filtration. 


By <Ref> (and <Ref>) the category BKF^∘ of Breuil-Kisin-Fargues modules up to isogeny is equivalent to the category of triples 

    (ℱ,ℱ^',β)
   
satisfying the same condition as in the statement of this theorem. Thus we are left with showing that the rigidifications can be identified under the equivalence in <Ref>. Let (M,φ) be a finite free Breuil-Kisin-Fargues module with corresponding quadruple (ℱ,ℱ^',β,T) as in <Ref>. Let N be the φ-module over B^+_ associated with ℱ^' in <Ref>. As was shown in the proof of <Ref> there exists a canonical isomorphism

    M⊗_A_infB^+_≅ N.

Recall that we have fixed a splitting k→𝒪_C/p of the projection 𝒪_C/p→ k.
Let

    ℰ(-)φ-Mod_W(k)[1/p]→Bun_X_FF

be the exact tensor functor associated with this splitting (cf. <cit.>). Then for the vector bundle ℱ^' there is a canonical isomorphism

    ⊕_λ∈gr^λ(HN(ℱ^')≅ℰ(N⊗_B^+_W(k)[1/p]

and we see (writing M:=M⊗_A_infW(k)[1/p]≅ N⊗_B^+_W(k)[1/p]) that the set of isomorphisms

    α_M M⊗_A_infB^+_≅M⊗_W(k)[1/p]B^+_

is in bijection with the set of isomorphisms

    αℱ^'≅⊕_λ∈gr^λ(HN(ℱ^').

Clearly, the condition on α_M resp. α to reduce to the identity over W(k)[1/p] resp. the graded pieces of the Harder-Narasimhan filtration is preserved.
This finishes the proof.


We note that we actually proved that for a finite free Breuil-Kisin-Fargues module (M,φ_M) with corresponding modification 

    (ℱ,ℱ^',αℱ_|X_FF∖{∞}≅ℱ^'_|X_FF∖{∞},T⊆ H^0(X_FF,ℱ))
 
as in <Ref> there is a bijection

    {rigidifications M⊗_A_infB^+_≅ (M⊗_A_infW(k))⊗_W(k)B^+_}
    ≅{rigidifications ℱ^'≅⊕_λ∈gr^λ(HN(ℱ^'))}

where we call an isomorphism ℱ^'≅⊕_λ∈gr^λ(HN(ℱ^')) reducing to the identity on the graded pieces of the Harder-Narasimhan filtration a rigidification of the data (ℱ,ℱ^', α,T) or just of ℱ^'.
In particular, if ℱ^' happens to be semistable there exists a unique rigidification on (M,φ_M), namely the one given by the identity on ℱ^'.

Using <Ref> it is possible to give an alternative proof of <Ref>. Namely, the category of semistable vector bundles of a fixed slope is abelian and therefore the cokernel of a morphism between vector bundles on the Fargues-Fontaine curve which respects fixed splittings of the respective Harder-Narasimhan filtrations is again a vector bundle. This proves (using <Ref>) that the category BKF^∘_rig is abelian. Moreover, as it is a rigid tensor category, the existence of a fiber functor (cf. <Ref>) or Deligne's criterion proves that it is moreover Tannakian.

<Ref> also proves that the category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny does not depend (up to canonical isomorphism) on the choice of the section k→𝒪_C/p.
















We do not know how to describe the category of rigidified Breuil-Kisin-Fargues modules up to isogeny only in terms of pairs (V,Ξ) where V is a finite-dimensional _p-vector space and Ξ⊆ V⊗__pB_ a B_^+-lattice. However, we can record the following.



Let f (M,φ_M,α_M)→ (M^',φ_M^',α_M^')∈BKF_rig be a morphism of finite free rigidified Breuil-Kisin-Fargues modules. Let g:(T,Ξ)→ (T^',Ξ^') be the associated morphism between pairs of a _p- and B^+_-lattice as in <Ref>. Then the cokernel of the morphism

    ΞΞ^'
  
is free over B^+_.


By construction of (T,Ξ) resp. (T^',Ξ^') the morphism gΞ→Ξ^' is given by

    f⊗Id M⊗_A_infB^+_→ N⊗_A_infB^+_.

Set M:=M⊗_A_infW(k) resp. N:=N⊗_A_infW(k). By assumption the diagram

    M⊗_A_infB^+_[r]^f⊗Id[d]^α_M⊗ B^+_    N⊗_A_infB^+_[d]^α_N⊗ B^+_
    M⊗_A_infB^+_[r]^f⊗Id   N⊗_A_infB^+_.

with both vertical morphisms isomorphisms commutes. In particular, the cokernel of g is free.


Let us list some fiber functors, i.e., “realizations”, of the Tannakian category BKF_rig^∘.

We recall the following lemma which is a direct corollary of the classification of vector bundles on the Fargues-Fontaine curve (cf. <cit.>).



The category of φ-modules over W(k)[1/p], i.e., the category of isocrystals over k, is equivalent to the category of -graded vector bundles ℰ=⊕_λ∈ℰ^λ such that ℰ^λ is semistable of slope λ.  


Cf. <cit.>.  




Let (M,φ_M) be a Breuil-Kisin-Fargues module. We define


  * its “étale realization” T:=(M⊗_A_infW(C^♭))^φ_M=1, a _p-module.

  * its “crystalline realization” D:=M⊗_A_infW(k), a W(k)-module equipped with the φ-semilinear isomorphism φ_D:=φ_M⊗ W(k)[1/p]:φ^∗(D[1/p])≅ D[1/p] after inverting p.

  * and its “de Rham realization” V:=M⊗_A_inf,θ𝒪_C, a 𝒪_C-module. 





The realizations from <Ref> define fiber functors


  * ω_étBKF^∘_rig→Vec__p

  * ω_BKF^∘_rig→φ-Mod_W(k)[1/p]

  * ω_BKF^∘_rig→Vec_C

 on the Tannakian category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny.


The functors are defined by sending a Breuil-Kisin-Fargues module to the respective realizations (with p inverted).
As each Breuil-Kisin-Fargues modules is isogenous to a finite free Breuil-Kisin-Fargues module all three functors are exact. The functors given by crystalline and de Rham realization are clearly tensor functor. To prove that also the étale realization ω_ét defines a tensor functor we remark that it factors canonically into the tensor functor

    BKF^∘_rig→φ-Mod_W(C^♭)[1/p], (M,φ_M)↦ (M⊗_A_infW(C^♭)[1/p],φ_M⊗Id),
 
to isocrystals over C^♭ and the tensor functor

    φ-Mod_W(C^♭)[1/p]→Vec__p

sending an isocrystal to its isoclinic part of slope 0.


Actually, the étale realization ω_ét⊗__pC, base changed to C, underlies a filtered fiber functor (cf. <cit.> and <cit.>) on the category BKF^∘_rig.
This nicely parallels the case of rational Hodge structures.



Let (M,φ_M) be a finite free Breuil-Kisin-Fargues modules with associated étale realization

    T:=(M⊗_A_infW(C^♭))^φ_M=1

 and B^+_-lattice 

    Ξ:=M⊗_A_infB^+_⊆ T⊗__pB_≅ M⊗_A_infB_.

Then we define the decreasing filtration Fil^j(T⊗__pC), j∈, on 

    T⊗__pC=T⊗__pB^+_/(T⊗__pξ B^+_)
 
by

    Fil^j(T⊗__pC):=Im(ξ^jΞ∩ T⊗__pB^+_→ T⊗__pC)
  
with j∈.


Written in a suitable basis e_1,…,e_n of T⊗__pB^+_ (over B^+_) there exist λ_1,…,λ_n∈ satisfying λ_1≥…≥λ_n such that the lattice Ξ⊆ T⊗__pB_ is generated by the ξ^λ_1e_1,…,ξ^λ_ne_n.
Then the filtration Fil^j(T⊗__pC) is given by the image of

    ⟨ξ^λ_1+je_1,…, ξ^λ_n+je_n⟩∩ T⊗__pB^+_

in T⊗__pC, i.e., the C-subspace generated by the residue classes e_i of the e_i such that λ_i+j≤ 0. 



The fiber functor

    ω_ét⊗__pCBKF^∘_rig→Vec_C

admits a canonical filtration induced by the filtration in <Ref>, i.e., the functor

    ωBKF^∘_rig→FilVec_C, M↦ (ω_ét⊗__pC, Fil^∙(ω_ét(M)⊗__pC)
 
is exact.


Writing down adapted bases as was done following <Ref> one sees that ω is a tensor functor.
As symmetric powers and exterior powers are kernels of projectors (in characteristic 0) and the category FilVec_C of filtered vector spaces is pseudo-abelian we see that ω commutes with symmetric and exterior powers. Moreover, ω commutes with duals.
Let 0→ M→ N→ Q→ 0 be an exact sequence in BKF^∘_rig. We have to proof that the sequence

    0→gr(ω(M))→gr(ω(N))→gr(ω(Q))→ 0

is exact. Considering dimensions and taking duals it suffices to prove that the left arrow gr(ω(M))→gr(ω(N)) is injective. This is equivalent to the statement that

    Λ^r(gr(ω(M)))→Λ^r(gr(ω(N)))

is non-zero where r:=rk(M). As ω commutes with exterior powers we may replace the morphism M→ N by Λ^r M→Λ^r N and assume that r=1.  By <Ref>
Let (V,Ξ) resp. (V^',Ξ^') be the associated pairs of a _p- and a B^+_-lattice. We identify V,Ξ with their images in V^' resp. Ξ^'. Tensoring with the dual of (V,Ξ) we can assume that Ξ=V⊗__pB^+_. Pick a generator v∈ V. Then the element v⊗ 1 is an element in Ξ^' and moreover, it is part of a basis of Ξ^' by <Ref>. Therefore we can, cf. the description following <Ref>, find an adapted basis v:=e_1,…, e_n of V^'⊗ B^+_ containing v such that Ξ^'=⟨ e_1,ξ^λ_2e_2,…,ξ^λ_ne_n⟩. From the concrete description given after <Ref> it follows that the image of v in gr^0(V^'⊗__pC) is non-zero. This finishes the proof.


We remark that sending a finite free Breuil-Kisin-Fargues modules M∈BKF^∘ (without fixing a rigidification) to the filtered vector space ω_ét(M)⊗__pC is not an exact operation. An counterexample is provided by the morphism

    (_p,ξ B^+_)→ (_p,B^+_).




Let ω_?∈{ω_ét, ω_, ω_} be one of the fiber functors in <Ref>. Then we set

    G_?:=Aut^⊗(ω_?)

as the group of tensor automorphisms of ω_?.    


We do not know how to describe the affine group schemes G_ét, G_ or G_ apart from their maximal torus quotients corresponding to the full subcategory of rigidified Breuil-Kisin-Fargues modules admitting complex multiplication.
We record the following general result about these group schemes.



The band of the Tannakian category BKF^∘_rig is connected. In particular, the group schemes G_ét, G_ and G_ are connected.  


It suffices to show that every object M∈BKR^∘_rig such that the smallest abelian subcategory ⟨ M⟩, i.e., the full subcategory of subquotients of a finite direct sum ⊕ M, is closed under the tensor product is a direct sum of the unit object.
Assume that there exists an object like this. Set V:=ω_ét(M)=(M⊗_A_infW(C^♭)^φ_M=1[1/p]. Then the filtration of <Ref> on the étale realization ω_ét(M)⊗__pC=V⊗__pC must be trivial. Set Ξ:=M⊗_A_infB^+_⊆ V⊗__pB_. In a suitable basis e_1,…,e_n of V⊗__pB^+_ we can write Ξ=⟨ξ^λ_1e_1,…, ξ^λ_ne_n⟩ with λ_1,…,λ_n∈. Then for j∈ 

    ξ^jΞ⊆ V⊗__pB^+_

is generated by the ξ^max{λ_i+j,0}e_i. As the filtration on V⊗__pC is trivial we can conclude that

    λ_i≤ 0

and

    λ_i+1>0

for all i. In other words, λ_i=0 for all i, i.e., Ξ=V⊗__pB^+_, which proves that M is a direct sum of the unit object of BKF^∘_rig.


We now calculate the Ext-groups

    Ext^1_BKF^∘_rig(A_inf{d},A_inf{d^'})

for d,d^'∈. In particular, as they will turn out to be non-zero we will be able to conclude that the category BKF^∘_rig of Breuil-Kisin-Fargues modules is not semisimple (and thus the group schemes G_ét,G_,G_ are not reductive).

We recall one more period ring, namely B_e. By definition,

    B_e:=H^0(X_FF,𝒪_X_FF)=B_^φ=1.

Moreover, there exists the “fundamental sequence of p-adic Hodge theory” involving B_e:

    0→_p→ B_e→ B_/(B_^+)→ 0

(cf. this follows from <cit.>).



Let d∈. Then 

    Ext^1_BKF^∘_rig(A_inf,A_inf{d})≅
    B_/t^d B^+_.



We use <Ref> and will classify triples (ℱ,ℱ^',β) fitting into a commutative diagramm with exact rows (where a dotted arrow means an isomorphism outside ∞∈ X_FF)

    0 [r]    𝒪_X_FF[r]@–>[d]^t^d   ℱ[r]@–>[d]^β   𝒪_X_FF[r][d]^=    0 
    
    0 [r]    𝒪_X_FF(d) [r]    ℱ^'[r]    𝒪_X_FF[r]     0

and isomorphisms

    αℱ^'≅gr^∙(HN(ℱ^'))

inducing the canonical rigidifications on 𝒪_X_FF(d) resp. 𝒪_X_FF.
First we note that this last requirement actually implies that 

    ℱ^'≅𝒪_X_FF(d)⊕𝒪_X_FF.

Moreover, 

    ℱ≅𝒪_X_FF⊕𝒪_X_FF.

In other words, an extension of A_inf by A_inf{d} is thus determined by an automorphism, preserving the factor 𝒪_X_FF(d),

    𝒪_X_FF(d)⊕𝒪_X_FF

restricting to the identity on both factors, i.e., by an element in

    Hom_X_FF(𝒪_X_FF,𝒪_X_FF(d))≅ (B^+_)^φ=p^d,

and an isomorphism

    β (𝒪_X_FF⊕𝒪_X_FF)_|X_FF∖{∞}≅ (𝒪_X_FF(d)⊕𝒪_X_FF)_|X_FF∖{∞}

on X_FF∖{∞} (which again must preserve the filtration and associated gradeds). 
We thus obtain a canonical surjection (of _p-vector spaces)

    Γ B_et^d⊕ (B_^+)^φ=p^d→Ext^1_BKF^∘_rig(A_inf,A_inf{d})

by sending (a,b)∈ B_et^d⊕ (B_^+)^φ=p^d to the quadruple

    (𝒪_X_FF⊕𝒪_X_FF, 𝒪_X_FF(d)⊕𝒪_X_FF,β=
    [ t^d   a;   0   1 ],α=
    [ 1 b; 0 1 ])

Assume that a pair (a,b) defines a trivial extension. Then there exists 

    c∈_p≅ H^0(X_FF,𝒪_X_FF)
 
and 

    c^'∈ (B^+_)^φ=p^d≅ H^0(X_FF,𝒪_X_FF(d))
 
such that

    [ t^d   a;   0   1 ][ 1 c; 0 1 ]
    =
    [   1 c^';   0   1 ][ t^d   0;   0   1 ]
    [ 1 b; 0 1 ][   1 c^';   0   1 ]
    =
    [ 1 0; 0 1 ].

In other words, 

    b=-c^'
 
and

    ct^d+a=c^'=-b.

For b∈ (B^+_)^φ=p^d the element b/t^d lies in B_e. Therefore the pair 

    (b/t^dt^d,b)∈Ker(Γ)

lies in the kernel of Γ.
In particular, we see that the morphism

    Γ_|B_et^d B_et^d→Ext^1_BKF^∘_rig(A_inf,A_inf{d})

is still surjective. Moreover, its kernel is given by _pt^d because Γ(a,0) is trivial if and only if a∈_pt^d.
The fundamental exact sequence implies therefore

    Ext^1_BKF^∘_rig(A_inf,A_inf{d})≅ B_et^d/_pt^d≅ B_/t^dB^+_.
 


Let 

    K:=Ker(Ext^1_BKF^∘(A_inf,A_inf{d})→Ext^1_X_FF(𝒪_X_FF,𝒪_X_FF(d)))

be the kernel of the natural map (in terms of modifications it sends (ℱ,ℱ^',β) to ℱ^'). 
Analyzing the proof of <Ref> we see that

    K≅ B_et^d/(B^+_)^φ=p^d.
 
Thus imposing the condition of a rigidification enlarges this group to

    B_et^d/_pt^d≅ B_/t^dB^+_.


For more general Ext-groups we can prove the following result (which implies <Ref>).
For simplicity, we denote by 

    X^∘_FF:=X_FF∖{∞}=Spec(B_e)

the punctured Fargues-Fontaine curve.



Let (M_1,φ_M_1),(M_2,φ_M_2)∈BKF^0_rig(M_1,M_2) be rigidified Breuil-Kisin-Fargues modules up to isogeny with associated modifications 

    (ℱ_i,ℱ_i^',β_i,α_i), i=1,2,
 
of vector bundles on the Fargues-Fontaine curve as in <Ref>. Then there exist a natural surjection

    H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')↠Ext^1_BKF^∘_rig(M_1,M_2).

If ℱ_i^', i=1,2, are semistable, then the kernel of this surjection is naturally isomorphic to

    H^0(X_FF,ℱ_1^∨⊗_𝒪_X_FFℱ_2).



We first construct a natural morphism

    Γ H^0(X_FF^∘,ℱ_1|X^∘_FF^'∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')→Ext^1_BKF^∘_rig(M_1,M_2).
  
Let 

    b∈ H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')≅Hom_X_FF^∘(ℱ_1|X^∘_FF,ℱ_2|X^∘_FF^')
 
be an element. Then we set Γ(b) to be the extension

    0 [r]    ℱ_2[r]@–>[d]^β_2   ℱ_2⊕ℱ_1 [r]@–>[d]^[ β_2   b;   0 β_1 ]   ℱ_1[r]@–>[d]^β_1    0 
    
    0 [r]    ℱ_2^'[r]    ℱ^'_2⊕ℱ^'_1 [r]    ℱ_1^'[r]     0

with rigidification

    [ α_2   0;   0 α_1 ]ℱ^'_2⊕ℱ^'_1≅gr^∙(HN(ℱ^'_2⊕ℱ^'_1))=gr^∙(HN(ℱ^'_2))⊕gr^∙(HN(ℱ^'_1)).

We claim that Γ is surjective with kernel, if ℱ_1^',ℱ_2^' are semistable, given by 

    H^0(X_FF,ℱ_1^∨⊗_𝒪_X_FFℱ_2)≅Hom_X_FF(ℱ_1,ℱ_2),

embedded into H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^') via c↦β_2∘ c_|X^∘_FF.
Let 

    (ℱ,ℱ^',β,α)

be an extension of (ℱ_1,ℱ_1^',β_1,α_1) by (ℱ_2,ℱ_2^',β_2,α_2). Then 

    ℱ≅ℱ_1⊕ℱ_2

because H^1(X_FF,𝒪_X_FF)=0.
Moreover, as the extension must be compatible with the rigidification the extension

    0→ℱ_2^'→ℱ^'→ℱ^'_1→ 0

splits, i.e., ℱ^'≅ℱ^'_1⊕ℱ^'_2. We moreover see that the rigidification α must be given by a matrix

    α=[ α_2   a;   0 α_1 ]

with a∈Hom_X_FF(ℱ^'_1,gr(HN(ℱ^'_2))). Moreover, a must map to zero on the associated graded of the Harder-Narasimhan filtration

    gr(a)gr(HN(ℱ^'_1))→gr(HN(ℱ^'_2))
 
as α must reduce to the identity on these graded pieces.
Therefore there exists an a^'∈Hom_X_FF(ℱ^'_1,ℱ^'_2) such that

    α_2∘ a^' = gr(a^')∘α_1 + a.
 
Then the extension

    (ℱ,ℱ^',β,α)

is isomorphic to

    (ℱ,ℱ^',
    [   1 a^';   0   1 ]∘β,
    [ α_2   0;   0 α_1 ]
    )

and thus this extension lies in the image of Γ. Now assume that for 

    b∈ H^0(X_FF^∘,ℱ_1|X^∘_FF^∨⊗_𝒪_X^∘_FFℱ_2|X^∘_FF^')
 
the extension 

    Γ(b)

is trivial. This means that there exists two isomorphisms 

    [   1 c_1;   0   1 ]ℱ_1⊕ℱ_2≅ℱ_1⊕ℱ_2,

with c_1∈Hom_X_FF(ℱ_1,ℱ_2), and

    [   1 c_2;   0   1 ]ℱ^'_1⊕ℱ^'_2≅ℱ^'_1⊕ℱ^'_2,

with c_2∈Hom_X_FF(ℱ_1^',ℱ_2^') inducing zero on the graded pieces of the Harder-Narasimhan filtration, such that

    [ β_2   b;   0 β_1 ][   1 c_1;   0   1 ]=
    [   1 c_2;   0   1 ][ β_2   0;   0 β_1 ].

If ℱ^'_i,i=1,2, are semistable, then c_2=0 must be zero and thus 

    b+β_2∘ c_1=0,

i.e., b=-β_2∘ c_1. This implies therefore the last statement in the lemma.


We want to deduce that the category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is of homological dimension 1.



The category BKF^∘_rig of rigidified Breuil-Kisin-Fargues modules up to isogeny is of homological dimension 1, i.e., for every (M,φ_M), (N,φ_N)∈BKF^∘_rig the Ext-group (in the sense of Yoneda or equivalently as spaces of homomorphisms in the derived category) vanishes for i≥ 2, i.e., 

    Ext^i_BKF^∘_rig(M,N)=0.



<Ref> implies that the functor 
Ext^1_BKF^∘_rig(M,-)  
preserves surjections of rigidified Breuil-Kisin-Fargues modules. Namely, for a surjection

    (N_1,φ_N_1)↠ (N_2,φ_N_2)

of rigidified Breuil-Kisin-Fargues modules the associated surjection

    ℱ_1^'↠ℱ_2^'

of vector bundles on the Fargues-Fontaine curve must be split due to the preservation of the given rigidifications.
For an abelian category with enough injectives (or projectives) this would finish the proof by embedding an object into an injective object and using the associated long exact sequence. However, the category BKF^∘_rig does not contain enough injectives (or projectives), hence we have to work a bit more.
For this let 𝒜 be an essentially small[This means that the isomorphism classes in 𝒜 form a set.] abelian category such that 

    Ext^1_𝒜(A,-)

preserves surjections for every A∈𝒜.
Consider the (fully faithful and exact) embedding

    𝒜→Ind(𝒜)

of 𝒜 into the category Ind(𝒜) of its Ind-objects. The category Ind(𝒜) need not have to have enough injectives in general (<cit.>), but it has if 𝒜 is essentially small which we assumed (in fact it is then Grothendieck abelian, cf. <cit.>). Hence, we can conclude that (cf. <cit.>)

    _A^'→ B Ext^k_𝒜(A,A^')≅Ext^k_Ind(𝒜)(A,B)

for A∈𝒜, B∈Ind(𝒜) where the colimit is running over all A^'∈𝒜 with a morphism to B.
Now let f B^'→ B^'' be a surjection of Ind-objects. Then f can be written as a filtered colimit of surjections A^'→ A^'' with A^',A^''∈𝒜 (cf. <cit.>). Using the above formular for the Ext-groups and our assumption on 𝒜 we can conclude that for A∈𝒜 the functor

    Ext^1_Ind(𝒜)(A,-)

preserves surjections. Using embeddings into injectives (of which there are enough in Ind(𝒜) by our assumption on essential smallness) we can conclude that 

    Ext^i_Ind(𝒜(A,-)=0

for A∈𝒜 and i≥ 2.
Using again the above formula for the Ext-groups we can conclude

    Ext^i_𝒜(A,A^')=0

for A,A^'∈𝒜 and i≥ 2.


Applying these considerations to the category 𝒜=BKF^∘_rig proves the lemma if we can show that the category BKF^∘_rig is essentially small. But by definition Breuil-Kisin-Fargues modules are finitely presented and the isomorphism classes of finitely presented A_inf-modules form a set. As the possibilities for adding a Frobenius or a rigidification form a set, we can conclude that the isomorphism classes of Breuil-Kisin-Fargues modules form a set as required.




§ CM BREUIL-KISIN-FARGUES MODULES



In this section we want to apply the formal CM theory of <Ref> to the case 

    𝒯:=BKF^∘_rig
 
of rigidified Breuil-Kisin-Fargues modules up to isogeny (cf. <Ref>).

Using Fargues' theorem <Ref> the classification of CM Breuil-Kisin-Fargues modules, i.e., CM objects in the Tannakian category 𝒯 (cf. <Ref>), is actually very simple - they are uniquely determined by their CM type (E,Φ) (cf. <Ref> and <Ref>).

Actually, we can prove a stronger integral statement. For this let E/_p be a commutative semisimple algebra and let 𝒪 be an order in E.


 A finite free Breuil-Kisin-Fargues module with CM by 𝒪 will mean a finite free Breuil-Kisin-Fargues module (M,φ_M) together with an injection 𝒪→End_BKF((M,φ_M)), such that rk(M)=rk__p(𝒪).  




Let E be a commutative, semisimple, finite-dimensional algebra over _p and let 𝒪⊆ E be an order in E. Then there is a natural equivalence of categories between finite free Breuil-Kisin-Fargues module with CM by 𝒪
and pairs (T,Φ) where T is a faithful 𝒪-module, finite free over _p of rank __p(𝒪), and functions ΦHom__p(E,C)→.
In particular, if 𝒪=𝒪_E is the maximal order, then there is a bijection of finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E (up to isomorphism) and “types” ΦHom__p(E,C)→.


By <Ref> finite free Breuil-Kisin-Fargues modules with CM by 𝒪 are equivalent to pairs (T,Ξ) with T as in the statement of this lemma and Ξ⊆ T⊗__pB_ a B^+_-sublattice, stable under 𝒪⊗__pB^+_.
But B^+_ contains _p, hence

    𝒪⊗__pB_≅ (E⊗__p_p)⊗__pB_≅∏_Hom__p(E,_p)B_
  
and a 𝒪⊗__pB^+_-stable sublattice will be uniquely determined by the valuation in each factor, i.e., by a function

    ΦHom__p(E,_p)=Hom__p(E,C)→.

If 𝒪=𝒪_E is the maximal order, then moreover every faithful 𝒪_E-module T which is finite free of rank d over _p must be isomorphic to 𝒪_E. This finishes the proof.


We remark that in general there are non-trivial examples of 𝒪-modules T satisfying the hypothesis in <Ref>.
We thank Bhargav Bhatt and Sebastian Posur for discussions about this point.
In general, one can take 𝒪≠𝒪_E and T:=𝒪_E. But there exist also less pathological examples. For example, let E=_p(p^1/4) and set 𝒪:=_p[p^2/4,p^3/4]. Then 𝒪/p≅_p[t^2,t^3]/t^4≅_p[x,y]/(x^2,xy,y^2) is not Gorentstein. In particular, 𝒪 is not Gorenstein as well (the dualizing complex commutes with (derived) base change). However, it is still Cohen-Macaulay. Hence, the dualizing module T:=ω_𝒪≅Hom__p(𝒪,_p) (cf. <cit.>) for 𝒪 yields an example. It is finite free over _p of rank rk__p𝒪 and End_𝒪(ω_𝒪)≅𝒪. As 𝒪 is not Gorenstein, ω_𝒪≠𝒪.   

We now analyse rigidifications of Breuil-Kisin-Fargues modules with CM.



Let E be a commutative semisimple algebra over _p of dimension d.
Let (M,φ_M) be a finite free Breuil-Kisin-Fargues module of rank rk(M)=d with an injection E↪End_BKF^∘((M,φ_M)). Then there exists a unique rigidification

    α M⊗_A_infB^+_≅ (M⊗_Ainf W(k))⊗_W(k)B^+_
  
which is preserved by E, i.e., E-linear.


We use <Ref> (respectively the remark following it) to argue with the modification (ℱ,ℱ^',β) of vector bundles on the Fargues-Fontaine curve associated with (M,φ_M). The algebra E acts by assumption on the vector bundle ℱ^' of rank d and we must produce a unique E-linear isomorphism

    ℱ^'≅gr^∙(HN(ℱ^')).
   
Decomposing E (and then ℱ^' accordingly) into factors reduces to the case that E is a field. Then we claim that ℱ^' must be semistable. Indeed, each subbundle ℰ^λ⊆ℱ^' in the Harder-Narasimhan filtration of ℱ^' must be stable under E. Let K be the function field of X_FF. Then E⊗__pK is again a field (because X_FF⊗__pE is again integral, cf. <cit.>) and passing to the generic point η∈ X_FF yields a E⊗__pK-stable flag in the 1-dimensional E⊗__pK-vector space ℱ^'_η. Hence, this flag is trivial and thus ℱ^' semistable.
This implies finally that there exists a unique E-linear rigidification for ℱ^', namely the identity of ℱ^'.


In other words, we can write down all rigidified Breuil-Kisin-Fargues modules up to isogeny with CM by a finite-dimensional commutative, semisimple _p-algebra E in terms of pairs (V,Ξ). Namely, V must be isomorphic (as an E-module) to E and Ξ⊆ E⊗__pB_ can be constructed explicitly from the type

    ΦHom__p(E,C)→.
 
We now want to write down the Breuil-Kisin-Fargues modules (M,φ_M) corresponding to a pair (E,Φ). This will require more work.

We note that for a finite free Breuil-Kisin-Fargues module (M,φ_M_Φ) as in <Ref> the function ΦHom__p(E,C)→ in <Ref> is precisely the type of the rigidified Breuil-Kisin-Fargues modules up to isogeny (_p⊗__pM,φ_M_Φ)∈BKF^∘_rig with respect to the filtered fiber functor

    ω_ét⊗__pCBKF^∘_rig→Vec_C

(cf. <Ref> and <Ref>). Namely, this follows from the concrete description of the filtration on ω_ét⊗__pC (cf. the discussion after <Ref> and the proof of <Ref>). 


In general a Breuil-Kisin-Fargues module with CM by a commutative semisimple _p-algebra E will decompose according to the factors of E. In particular, we may focus on the case where E is a field.
Hence, we fix a finite extension E of _p and denote by E_0⊆ E its maximal unramified subextension. We fix a uniformizer π∈ E. 

Let _p⊆ k be the algebraic closure of _p. By formal étaleness of _p over _p there exists a unique lifting  _p↪𝒪_C/p of the embedding _p→ k. Concretely, 

    _p=⋃_n≥ 0(𝒪_C/p)^φ^n=1.

Taking the inverse limit over Frobenius yields a canonical embedding

    _p↪𝒪_C^♭.

The algebraic closure of _p in _p⊗__pA_inf is contained in

    _p≅_p⊗__pW(_p)⊆_p⊗__pA_inf

where _p denotes the completion of the maximal unramified extension _p^un of _p. In particular, the algebraic closure of _p in _p⊗__pA_inf is given by _p^un.
For an embedding of ι E_0↪_p^un we define

    A_inf,𝒪_E,ι:=𝒪_E⊗_𝒪_E_0,ιA_inf,

the ring of “ramified Witt vectors” (cf. <cit.>). Its elements are formal power series

    ∑_i=0^∞ [x_i]π^i

with x_i∈𝒪_C^♭.
We set

    A_inf,E,ι:=_p⊗__pA_inf,𝒪_E,ι=E⊗_𝒪_EA_inf,𝒪_E,ι.





Let τ E→ C be an embedding and let τ_0 E_0→ C be its restriction to E_0. Then the kernel of the homomorphism

    θ_τ𝒪_E⊗__pA_inf→ C, e⊗ x↦τ(e)θ(x)
 
is principal, generated by a non-zero divisor ξ_τ.
In the decomposition

    𝒪_E⊗__pA_inf≅𝒪_E⊗_𝒪_E_0(𝒪_E_0⊗__pA_inf)≅∏_ι E_0↪ C A_inf,𝒪_E,ι

the element ξ_τ can be chosen to be

    (1,…,π-[τ(π)^♭],…,1)

with π-[τ(π)^♭] placed in the component τ_0.
Here τ(π)^♭=(τ(π),τ(π)^1/p,…) denotes a p-power compatible systems of p-power roots of τ(π)∈ C.


The morphism θ_τ(C)→(E⊗__pA_inf) must factor through one component and this component must be the one corresponding to the factor A_inf,E,τ_0 because θ_τ factors over A_inf,E,τ_0. Then the statement is well-known (cf. <cit.>).


Being a non-zero divisor the element ξ_τ in <Ref> is unique up to a unit.
Moreover, 

    ∏_τ∈Hom__p(E,C)ξ_τ=uξ

with u∈𝒪_E⊗__pA_inf a unit. Indeed, tensoring the exact sequence

    0→ A_infA_inf𝒪_C→ 0

with 𝒪_E yields the sequence

    0→𝒪_E⊗__pA_inf𝒪_E⊗__pA_inf∏_τ𝒪_E→𝒪_C𝒪_C≅𝒪_E⊗__p𝒪_C→ 0

which implies that the vanishing locus ξ and ∏_τξ_τ generate the same ideal. As both elements are non-zero divisors, they differ by a unit. 




Let Φ__p(E,C)→ be a type. Then we define the finite free Breuil-Kisin-Fargues module with CM by 𝒪_E as
M_Φ:=𝒪_E⊗__pA_inf with Frobenius

    φ_M_Φ:=ξ̃_Φφ

where

    ξ̃_Φ:=∏_τ∈Hom__p(E,C)φ(ξ_τ)^Φ(τ)

with ξ_τ as in <Ref> and φ=Id⊗φ_A_inf𝒪_E⊗__pA_inf→𝒪_E⊗__pA_inf.


First observe that

    ξ̃_τ:=φ(ξ_τ)

is a generator of the morphism

    θ̃_τ𝒪_E⊗__pA_inf→ C, e⊗ x↦τ(e)θ̃(x)

which extends the morphism θ̃ A_inf→ C with kernel ξ̃. Hence  every ξ̃_τ is a unit in 𝒪_E⊗__pA_inf[1/ξ̃] and thus 

    φ_M_Φφ^∗(M_Φ)[1/ξ̃]→ M_Φ[1/ξ̃]
 is indeed an isomorphism of A_inf[1/ξ̃]-modules.
Moreover, the multiplication by 𝒪_E induces a multiplication on M_Φ and thus M_Φ is a finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E. To determine the isomorphism class of M_Φ it thus suffices (cf. <Ref>) to compute the type of M_Φ (which of course will turn out to be Φ).
We check independently of <Ref> that the (isomorphism class) of the module (M_Φ,φ_M_Φ) is independent of the choice of the elements ξ_τ.

We recall the following lemma. If d is degree of E_0 over _p, then we set 

    φ_E_0:=φ^d.
 


Fix ι∈Hom__p(E_0,C). Then for every x=∑_i≥ 0^∞[x_i]π^i∈ A_inf,𝒪_E,ι with x_0≠ 0 the 𝒪_E-module 

    P_x:={y∈ A_inf,𝒪_E,ι | φ_E_0(y)=xy}

is free of rank 1. Furthermore, if x∈ A_inf,𝒪_E,ι is a unit, then a generator of P_x is a unit in A_inf,𝒪_E,ι.


After tensoring with _p the first assertion is proven in the proof of <cit.>. But P_x is ϖ-torsion free and hence free over 𝒪_E of rank 1.  The second part follows from the proof of <cit.>.


For example, if E=_p and x=ξ̃, then y=μ spans the space P_ξ̃ as

    φ(μ)=[ε^p]-1=[ε^p]-1/[ε]-1([ε^p]-1)=ξ̃μ.
 
In this case, the space P_ξ̃ does not contain a unit in A_inf.

We strengthen <Ref> a bit to handle the non-connected ring 𝒪_E⊗__pA_inf as well.



Let x∈𝒪_E⊗__pA_inf such that x maps to a unit in 𝒪_E/p⊗__pC^♭. Then the space

    P_x:={y∈𝒪_E⊗__pA_inf | φ(y)=xy}

is free of rank 1 over 𝒪_E.
Here φ denotes the Frobenius _𝒪_E⊗φ_A_inf on 𝒪_E⊗__pA_inf. If x is already a unit in 𝒪_E⊗__pA_inf, then P_x contains a unit in 𝒪_E,⊗__pA_inf.


Write 

    𝒪_E⊗__pA_inf≅∏_ι∈Hom__p(E_0,C) A_inf,𝒪_E,ι

and thus the element x=(x_ι)_ι∈Hom__p(E_0,C) accordingly. The assumption on x implies that each x_ι satisfies the assumption in <Ref>. If x∈𝒪_E⊗__pA_inf is a unit, then as well each x_ι is a unit. Moreover, the Frobenius φ permutes the factors cyclically. The power

    φ^d=φ_𝒪_E_0

fixes every factor and induces, for a fixed ι, the morphism φ_𝒪_E_0 on A_inf,𝒪_E,ι. Fix some ι_0 and let y_0∈ A_inf,𝒪_E,ι be a generating solution of

    φ_𝒪_E(y_0)=x_ι_0 y_0
 
(y_0 is a unit if x is a unit). As φ permutes the factors, it is clear that we get a generating solution y∈ A_inf,𝒪_E,ι (which is a unit if x is a unit) for the equation

    φ(y)=xy.

Moreover, this y must be unique up to multipliciation by 𝒪_E^×.




The Breuil-Kisin-Fargues module (M_Φ,φ_M_Φ) in <Ref> is up to isomorphism independent of the choice of the elements ξ_τ in <Ref>.


A different choice of the elements ξ_τ yields the Breuil-Kisin-Fargues module

    (M^'_Φ,φ_M^'_Φ):=(𝒪_E⊗__pA_inf,uξ̃_Φφ)

with u∈ (𝒪_E⊗__pA_inf)^× a unit.
By <Ref> we can find a unit

    y∈ (𝒪_E⊗__pA_inf)^×

such that

    φ(y)=uy.

Then multiplication by y will define an (𝒪_E-linear) isomorphism

    (M^'_Φ,φ_M^'_Φ)→ (M_Φ,φ_M_Φ).

  


We can now describe all (rigidified) Breuil-Kisin-Fargues modules with CM.



Let ΦHom__p(E,C)→ be a type. Then the finite free Breuil-Kisin-Fargues modules

    (M_Φ,φ_M_Φ)
 
from <Ref> has type Φ. Moreover, every finite free Breuil-Kisin-Fargues modules with CM by 𝒪_E with type Φ is isomorphic to (M_Φ,φ_M_Φ).


If we can show that (M_Φ,φ_M_Φ) has type Φ, then the last assertion follows from <Ref>.  
Fix as in <Ref> elements

    ξ_τ∈𝒪_E⊗__pA_inf

generating the kernel of

    θ_r𝒪_E⊗__pA_inf→𝒪_C

with τ running over all embeddings τ E→ C.
For simplicity we may (after multiplying one ξ_τ by a unit in 𝒪_E⊗__pA_inf) assume that

    ξ=∏_τξ_τ

(cf. the discussion after <Ref>).
Set

    ξ̃_τ:=φ(ξ_τ)∈𝒪_E⊗__pA_inf

and let μ_τ∈𝒪_E⊗__pA_inf be as in <Ref> the unique (up to multiplication by 𝒪_E^×) generator for  elements satisfying the equation

    φ(μ_τ)=ξ̃_τμ_τ

(if E=_p and τ_p→ C is the unique inclusion, then ξ_τ can be taken to be ξ, which implies μ_τ=μ (up to a unit)).
Note that μ_τ is a unit in E⊗__pA_inf[1/μ]. Indeed, the product

    ν:=∏_τμ_τ

satisfies

    φ(ν)=ξ̃ν

as we assumed ξ=∏_τξ_τ.
Hence, by <Ref> ν and μ differ by a scalar in E.
Applying φ^-1 to the defining equation of μ_τ yields 

    μ_τ=ξ_τφ^-1(μ_τ)
 
and we see that φ^-1(μ_τ) is a unit in (E⊗__pA_inf)⊗_A_infB^+_ because the same is true for φ^-1(μ) (one checks θ(φ^-1(μ))=ζ_p-1≠ 0).
We note that the type of M_Φ depends only on the Breuil-Kisin-Fargues modules up to isogeny E⊗_𝒪_EM_Φ.
Set 

    V:=(M_Φ⊗_A_infW(C^♭)[1/p])^φ_M_Φ=1

Then we know that V is a one-dimensional E-vector space. Recall that

    φ_M_Φ=ξ̃_Φφ

with 

    ξ̃_Φ:=∏_τξ̃_τ^Φ(τ).

We can explicitly find a generator of V, namely the element

    μ_Φ:=∏_τμ_τ^-Φ(τ)∈ E⊗__pA_inf[1/μ].

Indeed,

    φ(μ_Φ)=ξ̃_Φ^-1μ_Φ,

which implies

    φ_M_Φ(μ_Φ)=μ_Φ.

By definition, the Breuil-Kisin-Fargues modules (up to isogeny) (E⊗_𝒪_EM_Φ,φ_M_Φ) corresponds to the pair

    (V,Ξ:=M_Φ⊗_A_infB^+_⊆ V⊗__pB_)

in (the isogeny version) of <Ref>. 
Now, the E⊗__pB^+_-lattice 

    V⊗__pB^+_⊆ V⊗__pB_=E⊗__pB_
 
is generated by μ_Φ. But μ_Φ equals, up to a unit in E⊗__pB^+_, the inverse ξ_Φ^-1 of the element

    ξ_Φ:=∏_τξ_τ^Φ(τ)=φ^-1(ξ̃_Φ)

 in E⊗__pB^+_ while the E⊗__pB^+_-lattice Ξ is generated by 1∈ E⊗__pB^+_. In other words, we find that

    Ξ=ξ_Φ(V⊗__pB^+_)

which implies that (M_Φ,φ_M_Φ) has type Φ by looking at the explicit decomposition

    E⊗__pB_≅∏_τ∈Hom__p(E,C)B_

under which ξ_Φ maps to the element (ξ^Φ(τ))_τ.


We want to finish this chapter with a concrete description of the automorphisms

    Aut^⊗(ω_ét,CM)

of the fiber functor, “the étale realization”,

    ω_ét,CMBKF^∘_rig,CM→Vec__p

on Breuil-Kisin-Fargues modules with CM.

Recall the pro-torus 

    D__p:=__p/L/_pL^∗
 
which was introduced before <Ref>. 



The reflex norm (with respect to the “étale realization”)

    rBKF^∘_rig,CM→Rep__p(D__p)

is an isomorphism. In particular,

    Aut^⊗(ω_ét,CM)≅ D__p.



We want to use <Ref>. In <Ref> with have proven that every rigidified Breuil-Kisin-Fargues modules (up to isogeny) with trivial filtration on its étale realization is actually trivial, i.e., a direct sum of the unit object. This implies that the functor r is fully faithful by <Ref> and then <Ref> implies, using again <Ref>, that r is essential surjective. This finishes the proof.  


We remark that <Ref> implies that the category BKF^∘_rig,CM of rigidified Breuil-Kisin-Fargues modules up to isogeny admitting CM is generated by Breuil-Kisin-Fargues modules (M,φ_M) whose associated B_^+-lattice Ξ⊆ T⊗__pB_ is minuscule, i.e.,

    ξ (T⊗__pB^+_)⊆Ξ⊆ T⊗__pB^+_.
 
By <cit.> these Breuil-Kisin-Fargues modules are associated to p-divisible groups over 𝒪_C. In particular, we see that the category BKF^∘_rig,CM of CM Breuil-Kisin-Fargues modules is generated (as a tensor category) by Breuil-Kisin-Fargues modules associated with p-divisible groups admitting CM. This is analogous to the case of rational Hodge structures admitting CM (cf. <cit.>). 


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