\'Etale cohomology of Stein algebras

TL;DR

This paper establishes an isomorphism between singular and étale cohomology for Stein spaces, linking topological and algebraic properties of Stein algebras and spaces.

Contribution

It proves the equivalence of singular and étale cohomology for Stein spaces and explores the algebraic origin of cohomology classes.

Findings

Singular cohomology with finite coefficients is isomorphic to étale cohomology for Stein spaces.

Cohomology classes originate from algebraic varieties via holomorphic maps.

Cohomology classes vanish outside nowhere dense analytic subsets.

Abstract

We prove that the singular cohomology with finite coefficients of a finite-dimensional Stein space $S$ is isomorphic to the \'etale cohomology of the Stein algebra $\mathcal{O}(S)$. We deduce that any class in $H^k(S,\mathbb{Z})$ comes from an algebraic variety by pullback by a holomorphic map (if $k\geq 1$), and vanishes on the complement of a nowhere dense closed analytic subset of $S$ (if $k\geq 2$).