Alain Genestier, Vincent Lafforgue
In this article, we apply the methods of our work on Fontaine's theory in equal characteristics to the $φ/\mathfrak S$-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem "weakly admissible implies admissible" of Colmez-Fontaine.
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We define and study Harder-Narasimhan filtrations on Breuil-Kisin-Fargues modules and related objects relevant to p-adic Hodge theory.
For $K$ an extension of $\mathbb{Q}_{p}$ with ring of integers $R$ we show how Breuil-Kisin modules can be used to determine Hopf orders in $K$-Hopf algebras of $p$-power dimension. We find all cyclic Breuil-Kisin modules, and use them to compute all of the Hopf orders in the group ring $KΓ$ where $Γ$ is cyclic of order $p$ or $p^{2}.$ We also give a Laurent series interpretation of the Breuil-Kisin modules that give these Hopf orders.
For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
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".
In the work we have considered Breuil-Kisin module over the ring of witt vectors $W(κ)$ over the residue field $κ$ of characteristic $p$ and a finite flat $\mathbb{Z}_p$-algebra $R$. Then considered Breuil-Kisin modules $M$ over the ring $W(κ)$ and taking the action of $R$ on $W(κ)$, we get again a Breuil-Kisin module $M$ over the ring $R \otimes_{\mathbb{Z}_p} W(κ)$. We have studied freeness and projectiveness of this module.
Assume that $p>2$, and let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth formally finite-type $\mathscr{O}_K$-algebra. (Indeed, we allow slightly more general rings $R$.) We construct an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(φ,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). A similar classification result for $p$-power order finite flat group schemes is deduced from the classificat…
Breuil-Kisin modules and integral $p$-adic Hodge theory
本文通过构造带 Galois 作用的 Breuil-Kisin 模范畴，分类积分半稳定 Galois 表示，并给出有限 $E(u)$-高度表示的分类，从而建立积分 $p$-进 Hodge 理论的代数对应物。
The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups
该研究引入并研究带$\u0120$-结构与下降数据的Breuil-Kisin模的模叠，定义了权有界的对应模叠，证明其为$p$进形式代数叠且与扭Schubert簇光滑等价，是作者博士论文的整理版本。
Let p be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we prove the existence of higher canonical subgroups with expected properties for G if the Hodge height of G is less than 1/(p^{n-2}(p+1)), including the case of p=2.
Properties of Breuil-Kisin modules inherited by $p$-divisible groups
研究在有限平坦 $\mathbb{Z}_p$-代数忠实作用于 $p$-可除群时，如何构造相应的 Breuil-Kisin 模，并分析其自由性与投射性。
Let p>2 be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we show that an upper ramification subgroup G^j+ is free of rank d over Z/p^nZ if the Hasse invariant of G is less than 1/(2p^(n-1)). We also prove the usual properties as the canonical subgroup.
Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups
本文证明Breuil-Kisin函子在半稳定表示格上的正合性，并利用该结果通过强可分模范畴中的Ext群计算积分Bloch-Kato Selmer群，同时定义了与Galois表示函子交换的张量积，应用于阿贝尔簇的杯积映射分解。
We use the Taylor-Wiles-Kisin patching method to prove some new cases of the Breuil-Schneider conjecture.
On the Emerton-Gee stack of potentially semistable representations
该研究利用Breuil-Kisin模变体分类潜在半稳定伽罗瓦表示，构建其栈并证明与Emerton-Gee栈同构，构造相关映射并证明其光滑性，得到对应全局正则性结果。
Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
Let $K$ be a mixed characteristic complete discrete valuation field with residue field admitting a finite $p$-basis, and let $G_K$ be the Galois group. We first classify semi-stable representations of $G_K$ by weakly admissible filtered $(φ,N)$-modules with connections. We then construct a fully faithful functor from the category of \emph{integral} semi-stable representations of $G_K$ to the category of Breuil-Kisin $G_K$-modules. Using the integral theory, we classify $p$-divisible groups over the ring of integers of $K$ by minuscule Breuil-Kisin modules with connections.
Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme μ_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of μ_{p^n,K}, in such a way that the construction of Kummer groups and Breuil…
We define and study certain moduli stacks of modules equipped with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin-Breuil $Φ$-modules used by Kisin in his study of deformation spaces of local Galois representations. We also define a version of a rigid analytic period map for these spaces, we show how their local structure can be described in terms of "local models", and we show how Bruhat-Tits buildings can be used to study their special fibers.
Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(μ_{p^r})$. Using crystalline Dieudonné theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(φ,Γ)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonné crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by…
We investigate the maximal finite length submodule of the Breuil-Kisin prismatic cohomology of a smooth proper formal scheme over a p-adic ring of integers. This submodule governs pathology phenomena in integral p-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic p, and (2) kernel of the specialization map in p-adic étale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine-Laffaille, Fontaine-Messing…
Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(φ,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(φ,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargue…
We show that unit $\mathcal{O}_{F,X}^Λ$-modules of Emerton and Kisin provide an analogue of locally constant sheaves in the context of Böckle-Pink $Λ$-crystals. For example they form a tannakian category if the coefficient algebra $Λ$ is a field. Our results hold for a big class of coefficien algebras which includes Drinfeld rings, and for arbitary locally noetherian base schemes.
Integral filtered Sen theory and applications
本文提出积分塞恩理论及其应用，通过构造满足特定收缩性质的算子，研究了与prismatic F-晶体等相关的滤波器性质，并获得了关于Hodge滤波器的消散和torsion绑定结果。
Prismatic $F$-gauges and a result of T. Liu
本文利用栈论方法重新证明了 Tong Liu 关于积分 Hodge 滤链分次部分挠元控制的结果，并显式描述了一般 p 进扩张的 Nygaard 栈中 Hodge–Tate 轨迹。
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings's almost purity theorem, along with a certain functor $Lη$ on the derived cate…
Crystalline condition for $A_{\mathrm{inf}}$-cohomology and ramification bounds
本文针对 $p$-进形式概形，提出系列条件 ($\mathrm{Cr}_s$) 控制棱镜上同调上的伽罗瓦作用，证明晶体性并推广分歧界。
In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $AΩ$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $AΩ$-complex by giving a cohomological construction o…
Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice
本文定义了p进Hodge结构的Tannakian范畴，并展示了其与可接受对之间的等价关系，通过研究p进周期的超越性来推广一维形式群的结果。
An integral analogue of Fontaine's crystalline functor
研究在完美域\(k\)上维特向量\(W\)上的光滑形式概型\(\mathfrak{X}\)，构造函子\(\mathbb{D}_{\mathrm{crys}}\)，证明其在特定条件下的性质及诱导范畴等价关系，还用它加强\(p -\)可除群的棱柱形迪厄多内理论研究。
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