Twisted differential operators of negative level and prismatic crystals

TL;DR

This paper develops a twisted differential calculus framework at negative levels, establishing an equivalence between certain modules via Frobenius pullback and connecting it to prismatic crystals and Cartier operators.

Contribution

It introduces a new twisted differential calculus at negative levels and proves a descent theorem linking modules with twisted connections through Frobenius pullback.

Findings

Frobenius pullback induces an equivalence between modules with twisted connections of levels -1 and 0.

Establishes a connection between twisted differential calculus and prismatic crystals.

Provides a foundation for further exploration of negative level differential operators.

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 quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.