Notions of Stein spaces in non-archimedean geometry

TL;DR

This paper establishes equivalent conditions characterizing Stein spaces in non-archimedean geometry, linking cohomological properties, complex-analytic notions, and exhaustion by compact domains, extending classical complex analysis concepts to the non-archimedean setting.

Contribution

It proves the equivalence of three different characterizations of Stein spaces in Berkovich's non-archimedean analytic geometry, including a new perspective involving exhaustion by compact domains.

Findings

Equivalence between cohomological vanishing and Stein properties.

Extension of Stein space characterization to spaces with boundary.

Connection between Liu's exhaustion domains and Kiehl's affinoid domains.

Abstract

Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension $k'$ of $k$, every coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When $X$ has no boundary the characterization is simpler: in~2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in 3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space. v2: major revision to handle also the case of spaces with boundary