Non-archimedean gauge seminorms

Abstract

This paper is intended to provide foundations to the theory of Witt-type topological group and ring functors defined on a category of topological algebras, and, in presence of Banach norms, to show how to topologically deal with them. It is logically the first of a series of papers in preparation on the use of Barsotti-Witt constructions to obtain Scholze's tilting equivalence uniformly with respect to the perfectoid field K of characteristic 0 lifting a particular perfectoid field F of characteristic p>0. The paper is basically self-contained and may have an independent interest especially for specialists of topological algebra and non-archimedean functional analysis: this accounts for its independent submission. We indicate a new viewpoint in the theory of non-archimedean Banach algebras, based on a higher-dimensional generalization of the notion of gauge-seminorm as explained in P. Schneider "Non-archimedean Functional Analysis" Springer 2002