Twisted differential operators of negative level and prismatic crystals

Abstract

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasinilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier transform on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.