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Syntomic complex and $p$-adic nearby cycles

Abhinandan

TL;DR

The paper develops a comprehensive framework linking local p-adic Hodge theory to global syntomic cohomology, showing that Galois cohomology of finite height crystalline representations can be computed via syntomic complexes with coherence in associated F-isocrystals. It builds a bridge between Wach and Fontaine–Laffaille theories and p-adic nearby cycles, through a network of equivalences between (phi,Gamma)-modules, overconvergent theories, and filtered de Rham data, culminating in a Fontaine–Messing–Lazard comparison that holds both locally and globally for schemes and p-adic formal schemes. Central contributions include precise p-adic quasi-isomorphisms between syntomic complexes with coefficients and continuous Galois cohomology, a detailed Poincaré lemma framework in fat/overconvergent settings, and global descent results that connect crystalline data to nearby cycles via filtered crystals and FM period maps. The work provides a robust, coefficient-sensitive generalization of the near-by cycles theorems, with explicit control of p-adic error terms and radii of convergence, enabling global comparisons for Fontaine–Laffaille modules and related p-adic sheaves. This has potential applications to global p-adic Hodge theory, integral p-adic cohomology, and arithmetic geometry over mixed characteristic bases.

Abstract

In local relative $p$-adic Hodge theory, we show that the Galois cohomology of a finite height crystalline representation (up to a twist) is essentially computed via the (Fontaine--Messing) syntomic complex with coefficients in the associated $F$-isocrystal. In global applications, for smooth ($p$-adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine--Laffaille module and the $p$-adic nearby cycles of the associated étale local system on the (rigid) generic fibre.

Syntomic complex and $p$-adic nearby cycles

TL;DR

The paper develops a comprehensive framework linking local p-adic Hodge theory to global syntomic cohomology, showing that Galois cohomology of finite height crystalline representations can be computed via syntomic complexes with coherence in associated F-isocrystals. It builds a bridge between Wach and Fontaine–Laffaille theories and p-adic nearby cycles, through a network of equivalences between (phi,Gamma)-modules, overconvergent theories, and filtered de Rham data, culminating in a Fontaine–Messing–Lazard comparison that holds both locally and globally for schemes and p-adic formal schemes. Central contributions include precise p-adic quasi-isomorphisms between syntomic complexes with coefficients and continuous Galois cohomology, a detailed Poincaré lemma framework in fat/overconvergent settings, and global descent results that connect crystalline data to nearby cycles via filtered crystals and FM period maps. The work provides a robust, coefficient-sensitive generalization of the near-by cycles theorems, with explicit control of p-adic error terms and radii of convergence, enabling global comparisons for Fontaine–Laffaille modules and related p-adic sheaves. This has potential applications to global p-adic Hodge theory, integral p-adic cohomology, and arithmetic geometry over mixed characteristic bases.

Abstract

In local relative -adic Hodge theory, we show that the Galois cohomology of a finite height crystalline representation (up to a twist) is essentially computed via the (Fontaine--Messing) syntomic complex with coefficients in the associated -isocrystal. In global applications, for smooth (-adic formal) schemes, we establish a comparison between the syntomic complex with coefficients in a locally free Fontaine--Laffaille module and the -adic nearby cycles of the associated étale local system on the (rigid) generic fibre.
Paper Structure (65 sections, 84 theorems, 104 equations)

This paper contains 65 sections, 84 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. - nearby cycles
  3. Finite height representations
  4. Syntomic coefficients and --modules
  5. Fontaine--Laffaille modules and nearby cycles
  6. Outline of the paper
  7. Relative --adic Hodge theory
  8. Setup and notations
  9. Period rings
  10. Fundamental exact sequence
  11. --adic Galois representations
  12. --modules
  13. The Galois group -
  14. Étale --modules
  15. Overconvergent étale --modules
  16. ...and 50 more sections

Key Result

Theorem 1.1

For $0 \leqslant k \leqslant r$, the natural map is a $p^N\textrm{-isomorphism}$, i.e. its kernel and cokernel are killed by $p^N$, where $N = N(e, p, r) \in \mathbb{N}$ depends on the absolute ramification index $e$ of $K$, prime $p$ and twist $r$ but not on $X$ or $n$.

Theorems & Definitions (217)

  • Theorem 1.1: colmez-niziol-nearby-cycles
  • Theorem 1.2: colmez-niziol-nearby-cycles
  • Theorem 1.3: abhinandan-crystalline-wach
  • Definition 1.4
  • Theorem 1.5: Theorem \ref{['thm:syntomic_complex_galois_cohomology']}
  • Corollary 1.6: Corollary \ref{['cor:syntomic_complex_galois_cohomology']}
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: Theorem \ref{['thm:lazard_fmlocal_comparison']}
  • ...and 207 more