Syntomic complexes and p-adic nearby cycles

TL;DR

This paper establishes a comparison between syntomic cohomology and p-adic nearby cycles for semistable schemes, extending previous results and proving a semistable conjecture using advanced cohomological techniques.

Contribution

It generalizes existing comparison theorems to broader settings and proves a semistable conjecture by linking syntomic cohomology with p-adic nearby cycles in semistable cases.

Findings

Established a comparison isomorphism up to constants for semistable schemes.

Extended comparison results to cases involving universal constants.

Proved a semistable conjecture for formal schemes with semistable reduction.

Abstract

We compute syntomic cohomology of semistable affinoids in terms of cohomology of $(\varphi,\Gamma)$-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of $p$-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finitness of \'etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.