\'Etale cohomology of Stein algebras

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 $k\geq 2$).