Local cyclic homology for nonarchimedean Banach algebras

TL;DR

This paper introduces local cyclic homology as a new invariant for nonarchimedean Banach algebras over a discrete valuation ring, demonstrating its key properties and relation to existing theories.

Contribution

It defines local cyclic homology for Banach V-algebras and proves its invariance under reduction mod p, homotopy, stability, and excision, advancing nonarchimedean algebraic analysis.

Findings

Local cyclic homology depends only on reduction mod p.

It is homotopy invariant.

It is matricially stable and excisive.

Abstract

Let V be a complete discrete valuation ring with uniformiser p. We introduce an invariant of Banach V-algebras called local cyclic homology. This invariant is related to analytic cyclic homology for complete, bornologically torsionfree V-algebras. It is shown that local cyclic homology only depends on the reduction mod p of a Banach V-algebra and that it is homotopy invariant, matricially stable, and excisive.