Dagger completions and bornological torsion-freeness

TL;DR

This paper introduces dagger algebras as bornological algebras over a discrete valuation ring with specific properties, explores their inheritance features, and describes their completions with examples.

Contribution

It defines dagger algebras with properties similar to Monsky-Washnitzer algebras and studies their completions and inheritance properties.

Findings

Dagger algebras are complete, bornologically torsion-free, and satisfy a spectral radius condition.

Inheritance properties of dagger algebra characteristics are analyzed.

Explicit dagger completions are described, including noncommutative examples.

Abstract

We define a dagger algebra as a bornological algebra over a discrete valuation ring with three properties that are typical of Monsky-Washnitzer algebras, namely, completeness, bornological torsion-freeness and a certain spectral radius condition. We study inheritance properties of the three properties that define a dagger algebra. We describe dagger completions of bornological algebras in general and compute some noncommutative examples.