Weak completions, bornologies and rigid cohomology

TL;DR

This paper clarifies the analysis of Monsky--Washnitzer completions using bornological algebra, leading to a functorial complex that computes rigid cohomology of algebraic varieties over fields of positive characteristic.

Contribution

It introduces a bornological approach to Monsky--Washnitzer completion, providing a functorial chain complex for computing rigid cohomology.

Findings

Provides a new bornological framework for completions

Constructs a functorial chain complex for rigid cohomology

Clarifies the analysis behind Monsky--Washnitzer completion

Abstract

Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.