Twisted calculus on affinoid algebras

TL;DR

This paper develops a framework for twisted differential operators on affinoid algebras, establishing an equivalence between differential and q-difference modules that preserves cohomological properties.

Contribution

It introduces twisted differential operators of a given radius, showing their independence from the endomorphism choice and linking differential systems to q-difference systems.

Findings

Established an explicit equivalence between differential and q-difference modules.

Proved the equivalence preserves cohomology and solutions.

Provided a new perspective on modules over affinoid algebras.

Abstract

We introduce the notion of a twisted differential operator of given radius relative to an endomorphism $$\sigma$$ of an affinoid algebra A. We show that this notion is essentially independent of the choice of the endomorphism $$\sigma$$. As a particular case, we obtain an explicit equivalence between modules endowed with a usual integrable connection (i.e. differential systems) and modules endowed with a $$\sigma$$-connection of the same radius (i.e. q-difference systems). Moreover, this equivalence preserves cohomology and in particular solutions.