Notions of Stein spaces in non-Archimedean geometry

Abstract

Let k k be a non-Archimedean complete valued field and let X X be a k 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 ′ k’ of  k k , every coherent sheaf on  X × k k ′ X \times _{k} k’ is acyclic; (2) X 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 X is compact); (3) X X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity.

When  X 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.