Breuil–Kisin Modules via Crystalline cohomology
Abstract.
For a perfect field of characteristic and a smooth and proper formal scheme over the ring of integers of a finite and totally ramified extension of , we propose a cohomological construction of the Breuil–Kisin modules attached to the adic étale cohomology . We then prove that our proposal works when , , and the crystalline cohomology of the special fiber of is torsionfree in degrees and .
Key words and phrases:
Breuil–Kisin Modules, crystalline cohomology2010 Mathematics Subject Classification:
Primary: 14F30 Secondary: 11F801. Introduction
Let be a perfect field of characteristic and a finite and totally ramified extension of . Fix an algebraic closure of and denote by its adic completion. If is a smooth proper formal scheme over with (rigid analytic) generic fiber , then the (torsion free part of the) adic étale cohomology is a stable lattice in a crystalline representation. Functorially associated to the linear dual is its Breuil–Kisin module over in the sense of^{1}^{1}1In fact, we work with a slightly different normalization than [KisinFcrystal], which is more closely related to (crystalline) cohomology; see Definition 3.1 and Remark 3.2 for details. [KisinFcrystal]. It is natural to ask for a direct cohomological construction of . While the work of Kisin [KisinFcrystal, 2.2.7, A.6] provides a link between Dieudonné crystals and Breuil–Kisin modules for Barsotti–Tate representations, this link amounts to a descent result from Breuil modules over dividedpower envelopes to Breuil–Kisin modules over , which is somewhat indirect (and limited to the case of Hodge–Tate weights and ). More recently, the work of Bhatt, Morrow, and Scholze [SBM] associates to any smooth and proper formal scheme over a perfect complex of modules whose cohomology groups are Breuil–Kisin–Fargues modules in the sense of [SBM, Def. 4.22] (see also Definition LABEL:BKFDef below), and which is an avatar of all integral adic cohomology groups of . One can deduce from their theory that if is defined over , then the base change is a Breuil–Kisin–Fargues module, and one has a canonical identification under the assumption that is torsionfree. We note that with this assumption, is also torsion free; see Theorem 14.5 and Proposition 4.34 of [SBM].
Unfortunately, this beautiful cohomological description of does not yield a cohomological interpretation of the original Breuil–Kisin module over , but only of its scalar extension to , which is a coarser invariant.
In this paper, assuming that is torsionfree for , we will provide a direct, cohomological construction of over , at least when . To describe our construction, we must first introduce some notation.
Fix a uniformizer of , and let be the minimal polynomial of over , normalized to have constant term . For each choose satisfying and define and . For we define , equipped with the unique continuous Frobenius endomorphism that acts on as the unique lift of the power map on and satisfies . We write for the continuous algebra surjection carrying to , and we view as a subring of by identifying ; this is compatible (via the ) with the canonical inclusions . We then see that is a (semilinear) isomorphism, so for the element
(1.1) 
makes sense in and, as a polynomial in , has zeroset the Galoisconjugates of . Define , so that for .
Write for the adic completion of the PDenvelope of , equipped with the adic topology. This is naturally a PDthickening of , equipped with a descending filtration obtained by taking the closure in of the usual PDfiltration. The inclusions uniquely extend to , and we henceforth consider as a subring of in this way. Note that uniquely extends to a continuous endomorphism which has image contained in (see Lemma 2.1). We identify and , and will frequently write .
Given a smooth and proper formal scheme over , we write for the base change to . As is a divided power thickening, we can then form the crystalline cohomology of relative to . It is naturally a finitetype module with a semilinear endomorphism and a descending and exhaustive filtration .
Give the localization the filtration by integral powers of , and equip with the tensor product filtration; that is, , with the sum ranging over all integers and taking place inside . We then define
We equip with the Frobenius map and define
We view as an module by .
With these preliminaries, we can now state our main result, which provides a cohomological description of Breuil–Kisin modules in Hodge–Tate weights at most :
Theorem 1.1.
Assume that . Let be a smooth and proper formal scheme over and an integer with , and let be the Breuil–Kisin module associated to the dual of the Galois lattice . If is torsionfree for , then there is a natural isomorphism of Breuil–Kisin modules
The proof of Theorem 1.1 has two major—and fairly independent—ingredients, one of which might be described as purely cohomological, and the other as purely (semi)linear algebraic. Fix a nonnegative integer , and let be the category of height quasiBreuil modules over , whose objects are triples where is a finite, free module, is a submodule containing with the property that is torsion free, and is a semilinear map whose image generates as an module. Morphisms are filtration and comaptible module homomorphisms. For each , we then define submodules for and we put for . We similarly define the category of height filtered Breuil–Kisin modules, whose objects are triples where is a finite and free module, is a submodule containing with having no torsion, and is a semilinear map whose image generates as an module, and we define for , with when . It is wellknown that is equivalent to the “usual” category of Breuil–Kisin modules over ; see Remark 3.2. Scalar extension along induces a covariant functor which is known to be an equivalence of categories [CarusoLiu, Theorem 2.2.1]. Our main “(semi)linearalgebraic” result is that the functor defined by is a quasiinverse to . This we establish using a structural result (Lemma LABEL:specialbasis) that provides an explicit description of a Breuil module via bases and matrices, together with a sequence of somewhat delicate Lemmas that rely on the fine properties of the rings and their endomorphisms .
On the other hand, if is a smooth and proper formal scheme and , then the crystalline cohomology can be naturally promoted to an object of . Using the results of Bhatt, Morrow, and Scholze [SBM], when is torsion free for , we prove in §LABEL:pdivGone that one has a canonical comparison isomorphism
from which we deduce that may be identified with the (filtered) Breuil–Kisin module attached to the linear dual of . Theorem 1.1 then follows.
2. Ringtheoretic constructions
We keep the notation of §1. Note that, by the very definition, the ring is topologically generated as an algebra by the divided powers . It follows that is the closure of the expanded ideal in . We write , which is a unit of . Since , one shows that for a unit . Observe that
by Legendre’s formula, so that the ring is naturally a subring of that contains and is stable under as . There are obvious inclusions that are compatible with the given inclusions and . By definition, the injective map has image precisely inside . While the naïve analogue of this fact for the rings is certainly false, Frobenius is nonetheless a “contraction” on in the following precise sense:
Lemma 2.1.
Let be a nonnegative integer and set . We have inside ; in particular, has image contained in . Moreover, if then for some and .
Proof.
Since is topologically generated as an module by and , to prove the first assertion it is enough to show that lies in for all . But this is clear, as and for all . To prove the second assertion, it likewise suffices to treat only the casees for . As observed above, for , so we compute
(2.1) 
Writing for the sum of the padic digits of any nonnegative integer and again invoking Legendre’s formula gives
which is nonnegative for . On the other hand, if then one has the inequality . Combining these observations with (2.1) then gives the desired decomposition with the sum of all terms in (2.1) with and the sum of the remaining terms. ∎
We now define
which—as is a ring homomorphism—has the natural structure of a ring via componentwise addition and multiplication. The fact that “contracts” the tower of rings manifests itself in the following Lemma, which inspired this paper:
Lemma 2.2.
The natural map
(2.2) 
is an isomorphism of rings.
Proof.
It is clear that the given map is an injective ring homomorphism, so it suffices to prove that it is surjective. Let be an arbitrary element of . Since , an easy induction using Lemma 2.1 shows that lies in , where is defined recursively by and for . As this holds for all and
so that is an increasing sequence (recall ), it follows that . But then is in the image of (2.2), as desired. ∎
For later use, we record here the following elementary result:
Lemma 2.3.
Let and be any nonnegative integers. Then

inside .

inside .
Proof.
We must prove that is injective with target that is torsion free. This is an easy induction on , using the fact that and are free of rank one over with generators and , respectively. ∎
3. Breuil and Breuil–Kisin modules
We begin by recalling the relation between Breuil–Kisin modules and Breuil modules in low Hodge–Tate weights. Throughout, we fix an integer
Definition 3.1.
We write for the category of height filtered Breuil–Kisin modules over whose objects are triples where:

is a finite free module,

is a submodule with and is torsion free.

is a semilinear map whose image generates as an module.
Morphisms are module homomorphisms which are compatible with the additional structures. For any object of and any , we put
(3.1) 
and set for , and define a semilinear map by the condition
(3.2) 
for . Note that for we have .
Remark 3.2.
Our definition of the category is perhaps nonstandard (cf. [CarusoLiu]). In the literature, one usually works instead with the category of Breuil–Kisin modules (without filtration), whose objects are pairs consisting of a finite free module and a semilinear map whose linearization is killed by , with evident morphisms. However, the assignment induces an equivalence between our category and the “usual” category of Breuil–Kisin modules . While this is fairly standard (e.g. [Lau:Frames, Lemma 8.2] or [VZ, Lemma 1]), for the convenience of the reader and for later reference, we describe a quasiinverse.
Given as above and writing for the linearization of , there is a unique (necessarily injective) linear map
The corresponding filtered Breuil–Kisin module over is then given by
(3.3) 
Alternatively, as one checks easily, we have the description
(3.4) 
From (3.3) it is clear that and are then free modules, so that has projective dimension over . It follows from the Auslander–Buchsbaum formula and Rees’ theorem that has depth as an module, so since has maximal ideal and is a zerodivisor on , it must be that is not a zero divisor on and hence this quotient is torsion free and we really do get a filtered Breuil–Kisin module in this way.
We have chosen to work with our category of filtered Breuil–Kisin modules instead of the “usual” category of Breuil–Kisin modules as it is our category whose objects are inherently “cohomological”, as we shall see.
Let be as above (adically completed PDenvelope of ), and for write for the (closure of the) ideal generated by . For One has , so we may define as .
Definition 3.3.
Denote by the category of height quasi Breuilmodules over . These are triples consisting of a finite free module with an submodule and a semilinear map such that:

and has no torsion.

The image of generates as an module
Morphisms are module homomorphisms that are compatible with the additional structures. Given a quasi Breuil module of height , for we set
(3.5) 
and we put for and define by the recipe
Note that on ; it follows that and determine each other.
There is a canonical “base change” functor
(3.6) 
defined as follows: if is an object of , then we define and , with the submodule generated by the images of and . Then by definition, the restriction of to has image contained in , so it makes sense to define on . Using the definition of the category , it is straightforward to check that this really does define a covariant functor from to .