On the sheafyness property of spectra of Banach rings

Abstract

Let be a non‐Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to a homotopical Huber spectrum via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces that is a homeomorphism in some well‐known examples of non‐sheafy Banach rings, where is the usual Huber spectrum of . This permits the use of the tools from derived geometry to understand the geometry of in cases when the classical structure sheaf is not a sheaf.