Cohomology of algebraic varieties over non-archimedean fields

TL;DR

This paper develops a sheaf cohomology theory for algebraic varieties over non-archimedean fields using Hrushovski-Loeser's stable completion, establishing foundational properties and linking it to Berkovich analytification.

Contribution

It introduces a new sheaf cohomology framework over non-archimedean fields and definable sets, connecting it with existing theories and proving key axioms and bounds.

Findings

Cohomology satisfies Eilenberg-Steenrod axioms

Finiteness and invariance of cohomology groups

Bounded cohomological dimension and finitely many types

Abstract

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma_\infty$, where $\Gamma$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.