| Résume | Abstract: In algebraic geometry, Serre's criterion for affineness provides a full characterisation of affine schemes in terms of vanishing higher cohomology. In non-Archimedean analytic geometry, however, such a cohomological criterion allows for more than just Berkovich spectra of affinoid algebras (the 'conventional local building blocks' of non-Archimedean analytic spaces). That is, in the late eighties, Qing Liu constructed explicit examples of non-affinoid compact non-Archimedean analytic spaces with vanishing higher cohomology. Consequently, compact non-Archimedean analytic spaces with vanishing higher cohomology are nowadays referred to as Liu spaces. More generally, when the space is is not necessarily compact, it is called a Stein space, inspired by the complex analytic setting. In this talk, I will report on recent work where I provide a way to fully describe Liu (resp. Stein) spaces as Berkovich spectra of certain Banach (resp. Fréchet) algebras. This answers a conjecture by Michael Temkin and can be interpreted as a non-Archimedean analytic version of Serre's criterion for affineness. |
