Big de Rham-Witt cohomology: basic results (Q462465)

Summary: 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.