Twisted calculus on affinoid algebras

We introduce the notion of a twisted differential operator of given radius relative to an endomorphism $σ$ of an affinoid algebra A. We show that this notion is essentially independent of the choice of the endomorphism $σ$. 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 $σ$-connection of the same radius (i.e. q-difference systems). Moreover, this equivalence preserves cohomology and in particular solutions.

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