Twisted calculus on affinoid algebras
Bernard Le Stum (IRMAR), Adolfo Quir\'os

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 of an affinoid algebra A. We show that this notion is essentially independent of the choice of the endomorphism . 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 -connection of the same radius (i.e. q-difference systems). Moreover, this equivalence preserves cohomology and in particular solutions.
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