On endomorphisms of the de Rham cohomology functor

TL;DR

This paper computes the moduli of endomorphisms of de Rham and crystalline cohomology functors for smooth schemes over truncated Witt vectors, leading to refined decompositions and functorial improvements in characteristic p>0.

Contribution

It provides a detailed computation of endomorphism moduli for these cohomology functors and applies the results to refine classical decompositions in positive characteristic.

Findings

Refined Drinfeld's decomposition for de Rham cohomology

Functorial improvements in cohomology decompositions

Enhanced understanding of endomorphisms in crystalline cohomology

Abstract

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the classical Deligne--Illusie decomposition result for de Rham cohomology of varieties in characteristic $p>0$ that are liftable to $W_2$, and prove further functorial improvements.