Big de Rham-Witt cohomology: basic results

Abstract

Let $X$ be a smooth projective $R$-scheme, where $R$ is a smooth $\mathbb{Z}$-algebra. As constructed by Hesselholt, we have the absolute big de Rham–Witt complex $\mathbb{W}{\Omega }_X^{\ast }$ of $X$ at our disposal. There is also a relative version $\mathbb{W}{\Omega }_{X/R}^{\ast }$ with $\mathbb{W}(R)$-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 ${\mathbb{W}}_S(R)$-module, provided that the de Rham cohomology is a flat $R$-module. In addition, we establish a Poincaré duality theorem. explicit description of the relative de Rham–Witt complex of a smooth $\lambda$-ring, which may be of independent interest.