Notions of Stein spaces in non-Archimedean geometry

Let $k$ be a non-Archimedean complete valued field and let $X$ 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.