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Crystalline part of the Galois cohomology of crystalline representations

Abhinandan

TL;DR

The paper develops a Wach-module–based approach to realize the crystalline part of local Galois cohomology via syntomic complexes. It defines two syntomic complexes, one over the period ring $A_F^+$ and a descended version over $S$, equipped with Nygaard filtrations and differential operators that lift the Bloch--Kato filtration. After inverting $p$, each complex computes the Bloch--Kato crystalline cohomology $H^k_f(G_F,V)$ for $k=0,1,2$, with the second degree vanishing in the AF$^+$ case; the constructions are compatible with the Fontaine--Herr framework and intimately connected to $D_{\mathrm{cris}}(V)$. A descent theorem for Wach modules is proved, showing an equivalence between Wach modules over $A_F^+$ and those over $S$, and yielding a coherent picture linking crystalline representations, $(\varphi,\Gamma)$-modules, and Wach modules. Collectively, the results provide a concrete, cohomological bridge from Wach-module data to Bloch--Kato Selmer groups, offering an explicit, integral perspective on local $p$-adic Hodge-theoretic invariants.

Abstract

For $p \geqslant 3$ and an unramified extension $F/\mathbb{Q}_p$ with perfect residue field, we define a syntomic complex with coefficients in a Wach module over a certain period ring for $F$. We show that our complex computes the crystalline part of the Galois cohomology (in the sense of Bloch and Kato) of the associated crystalline representation of the absolute Galois group of $F$. Furthermore, we establish that Wach modules of Berger naturally descend over to a smaller period ring studied by Fontaine and Wach. This enables us to define another syntomic complex with coefficients, and we show that its cohomology also computes the crystalline part of the Galois cohomology of the associated representation.

Crystalline part of the Galois cohomology of crystalline representations

TL;DR

The paper develops a Wach-module–based approach to realize the crystalline part of local Galois cohomology via syntomic complexes. It defines two syntomic complexes, one over the period ring and a descended version over , equipped with Nygaard filtrations and differential operators that lift the Bloch--Kato filtration. After inverting , each complex computes the Bloch--Kato crystalline cohomology for , with the second degree vanishing in the AF case; the constructions are compatible with the Fontaine--Herr framework and intimately connected to . A descent theorem for Wach modules is proved, showing an equivalence between Wach modules over and those over , and yielding a coherent picture linking crystalline representations, -modules, and Wach modules. Collectively, the results provide a concrete, cohomological bridge from Wach-module data to Bloch--Kato Selmer groups, offering an explicit, integral perspective on local -adic Hodge-theoretic invariants.

Abstract

For and an unramified extension with perfect residue field, we define a syntomic complex with coefficients in a Wach module over a certain period ring for . We show that our complex computes the crystalline part of the Galois cohomology (in the sense of Bloch and Kato) of the associated crystalline representation of the absolute Galois group of . Furthermore, we establish that Wach modules of Berger naturally descend over to a smaller period ring studied by Fontaine and Wach. This enables us to define another syntomic complex with coefficients, and we show that its cohomology also computes the crystalline part of the Galois cohomology of the associated representation.
Paper Structure (22 sections, 30 theorems, 75 equations)

This paper contains 22 sections, 30 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Bloch--Kato Selmer groups
  3. Syntomic complexes and Galois cohomology
  4. Syntomic complex over -
  5. Syntomic complex over -
  6. Outline of the paper
  7. - representations
  8. Period rings
  9. - representations
  10. Galois cohomology and Fontaine--Herr complex
  11. Bloch--Kato Selmer groups
  12. Wach modules
  13. Descent of Wach modules
  14. Wach modules and crystalline representations
  15. Nygaard filtration on Wach modules
  16. ...and 7 more sections

Key Result

Theorem 1.2

For each $k \in \mathbb{N}$, we have a natural isomorphism

Theorems & Definitions (81)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['thm:syntomic_blochkatoselmer_af+']}
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['thm:wachmod_s_af+_equiv']}
  • Definition 1.7
  • Theorem 1.8: Theorem \ref{['thm:syntomic_blochkatoselmer_s']}
  • Remark 1.9
  • Remark 1.10
  • ...and 71 more