On the de Rham–Witt complex in mixed characteristic

Abstract

The purpose of this paper is twofold. Firstly, it gives a thorough treatment of the de Rham–Witt complex for Z (p) -algebras, a construction we first considered in [L. Hesselholt, I. Madsen, Ann. of Math. 158 (2003) 1–113]. This complex is the natural generalization to Z (p) -algebras of the de Rham–Witt complex for F p -algebras of Bloch–Deligne–Illusie [L. Illusie, Ann. Sci. École Norm. Sup. 12 (4) (1979) 501–661] (for p odd). We also give an explicit formula for the de Rham–Witt complex of a polynomial ring in terms of that of the coefficient ring. Secondly, we generalize the main Theorem C of [L. Hesselholt, I. Madsen, Ann. of Math. 158 (2003) 1–113] to smooth algebras over a discrete valuation ring of mixed characteristic (0, p) with perfect residue field and p odd.