On the sheafyness property of spectra of Banach rings

Let (Formula presented.) 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 (Formula presented.) a homotopical Huber spectrum (Formula presented.) via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf (Formula presented.) 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 (Formula presented.) that is a homeomorphism in some well-known examples of non-sheafy Banach rings, where (Formula presented.) is the usual Huber spectrum of (Formula presented.). This permits the use of the tools from derived geometry to understand the geometry of (Formula presented.) in cases when the classical structure sheaf (Formula presented.) is not a sheaf.