Cohomology of locally-closed semi-algebraic subsets

TL;DR

This paper proves finiteness and cohomological properties of locally-closed semi-algebraic subsets in Berkovich spaces over non-archimedean fields, extending known results and providing new tools for non-archimedean geometry.

Contribution

It establishes finiteness of l-adic cohomology groups for semi-algebraic subsets in Berkovich spaces, generalizing previous results and including new cases in rigid analytic geometry.

Findings

Cohomology groups are finite-dimensional vector spaces.

Results apply to both Berkovich and adic spaces.

Finiteness holds in various characteristic settings.

Abstract

Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U}, Q_l) behave like H^i_c(\bar{X}, Q_l), where \bar{U} = U \otimes \hat{\bar{k}}. In particular, they are finite-dimensional vector spaces. This result has been used by E. Hrushovski and F. Loeser. Moreover, we prove analogous finiteness properties concerning rigid semi-analytic subsets of compact Berkovich spaces (resp. adic spaces associated to quasi-compact quasi-separated k-rigid spaces) when char(\tilde{k}) \neq 0 (resp in any characteristic).