Big de Rham-Witt cohomology: basic results

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 ∗ of X at our disposal. There is also a relative version W ∗ 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 WS(R)-module, provided that the de Rham cohomology is a flat R-module. In addition, we establish a Poincare duality theorem. explicit description of the relative de Rham-Witt complex of a smooth λ-ring, which may be of independent interest.