Rigid Cohomology and de Rham-Witt complexes

TL;DR

This paper demonstrates how rigid cohomology with compact supports for separated schemes over a perfect field of characteristic p can be computed using de Rham-Witt complexes, extending classical comparison theorems.

Contribution

It generalizes the Bloch-Illusie comparison theorem to include cohomologies relative to W_n with coefficients in flat crystals, broadening the scope of cohomological computations.

Findings

Rigid cohomology can be computed via de Rham-Witt complexes.

Extension of Bloch-Illusie theorem to relative cohomologies with coefficients.

Generalization applies to separated schemes of finite type over perfect fields.

Abstract

Let $k$ be a perfect field of characteristic $p > 0$, $W_n = W_n(k)$. For separated $k$-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the Bloch-Illusie theorem to the case of cohomologies relative to $W_n$ with coefficients in a crystal that is only supposed to be flat over $W_n$.