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Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

Pavel Čoupek, Evangelia Gazaki, Adriano Marmora

Abstract

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. The same statement is proved for Breuil-Kisin modules for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $δ_1\cupδ_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules.

Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

Abstract

Let be a -adic field and a lattice in a semistable representation of with Hodge-Tate weights in . Assuming , we prove that for a semistable extension of by , the corresponding sequence of strongly divisible modules is exact. The same statement is proved for Breuil-Kisin modules for all . In the crystalline case, we deduce that the integral Bloch-Kato Selmer group is computed by in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties over with good reduction, the cup product map induced by the Kummer sequences of factors through an group of strongly divisible modules.

Paper Structure

This paper contains 19 sections, 31 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Background and preliminary results
  3. Review of period rings
  4. The rings $\operatorname{\mathfrak{S}}$ and $\operatorname{A_{inf}}$
  5. The rings $S$, $A_{\operatorname{cris}}$ and $\widehat{A}_{\operatorname{st}}$
  6. Kisin modules and functors to $\mathbb{Z}_p$-lattices
  7. Breuil modules and functors to $\mathbb{Z}_p$-lattices
  8. The various categories of modules
  9. Functors to Galois representations
  10. Comparison functors between Kisin and Breuil modules
  11. Exactness Results
  12. Some counterexamples to exactness
  13. An exactness result for $1$-extensions
  14. An auxiliary exactness result
  15. Applications
  16. ...and 4 more sections

Key Result

Theorem A

Fix $r \in \mathbb{N}$ with $r<p-1$. Given a short exact sequence of lattices in semistable $G_K$-representations with Hodge-Tate weights in $[0, r]$ of the form \begin{tikzcd} 0\ar[r]& T \ar[r] & L \ar[r] & \ZZ_p \ar[r] & 0, \end{tikzcd}the sequence of strongly divisible modules \begin{tikzcd} 0\ar

Theorems & Definitions (92)

  • Theorem A: Corollary \ref{['cor:ExactBreuilModules']}, Theorem \ref{['Th:isomorphism_ext']}, Theorem \ref{['Th:isomorphism_ext_st']}
  • Theorem B: Theorem \ref{['thm:ExactKisinModules']}
  • Theorem C
  • Theorem D: Theorem \ref{['thm:cupproduct']}, Corollary \ref{['cor:abelianvars']}
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 82 more