On Tate Acyclicity and Uniformity of Berkovich Spectra and Adic Spectra

Abstract

We construct a non-sheafy uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. We also construct uniform affinoid rings in the sense of Roland Huber such that rational localisations of the adic spectra do not preserve the uniformity. One of them is an example of a non-sheafy uniform affinoid ring. We introduce the notion of local uniformity instead, and prove that the local uniformity implies the sheaf condition.