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 of X at our disposal. There is also a relative version \mathbb W\Omega^*_{X/R} 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.