Nonarchimedean bornologies, cyclic homology and rigid cohomology

TL;DR

This paper develops a new analytical framework using nonarchimedean bornologies to connect cyclic homology with rigid cohomology, providing a functorial chain complex for computing Berthelot's rigid cohomology.

Contribution

It introduces spectral radius estimates in bornological algebras to clarify Monsky--Washnitzer completion and constructs a chain complex linking cyclic homology to rigid cohomology.

Findings

Spectral radius estimates facilitate analysis of bornological algebras.

A functorial chain complex computes Berthelot's rigid cohomology.

Connection established between cyclic homology and rigid cohomology.

Abstract

Let $V$ be a complete discrete valuation ring with residue field $k$ and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using spectral radius estimates for bounded subsets in complete bornological $V$-algebras. This leads us to a functorial chain complex for commutative $k$-algebras that computes Berthelot's rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological $V$-algebras.