Cartier transform and prismatic crystals

TL;DR

This paper establishes a local equivalence between crystals on q-crystalline and prismatic sites via the Cartier transform, connecting it with the q-twisted Simpson correspondence, and ensuring compatibility with cohomology theories in dimension one.

Contribution

It explicitly identifies the Cartier transform with the q-twisted Simpson correspondence in a local setting, linking different categories of crystals.

Findings

Four compatible equivalences between categories are established.

The local identification simplifies understanding of the Cartier transform.

Compatibility with cohomology theories is demonstrated in dimension one.

Abstract

We show that the abstract equivalence of categories, called Cartier transform, between crystals on the q-crystalline and prismatic sites can be locally identified with the explicit local q-twisted Simpson correspondence. This establishes four equivalences that are all compatible with the relevant cohomology theories. We restrict ourselves for simplicity to the dimension one situation.