Big de Rham-Witt cohomology: basic results

TL;DR

This paper investigates the hypercohomology of the relative big de Rham-Witt complex for smooth projective schemes over smooth Z-algebras, establishing projectivity and a Poincaré duality under certain conditions.

Contribution

It proves the projectivity of hypercohomology modules of the relative big de Rham-Witt complex and establishes a Poincaré duality theorem for these complexes.

Findings

Hypercohomology modules are projective over Witt rings when de Rham cohomology is flat.

Establishment of a Poincaré duality theorem for the big de Rham-Witt complex.

Analysis of the hypercohomology of the complex after finite truncation.

Abstract

Let $X$ be a smooth projective $R$-scheme, where $R$ is a smooth $\Z$-algebra. As constructed by Hesselholt, we have the absolute big de Rham-Witt complex $\W\Omega^*_X$ of $X$ at our disposal. There is also a relative version $\W\Omega^*_{X/R}$ with $\W(R)$-linear differential. In this paper we study the hypercohomology of the relative (big) de Rham-Witt complex after truncation with finite truncation sets $S$. We show that it is a projective $\W_S(R)$-module, provided that the de Rham cohomology is a flat $R$-module. In addition, we establish a Poincar\'e duality theorem.