A geometric approach to the relative de Rham-Witt complex in the smooth, $\mathbb{Z}$-torsion free case

Abstract

Let $X$ be a smooth scheme over a finitely generated flat $\mathbb{Z}$-, $\mathbb{Z}_{(p)}$- or $\mathbb{Z}_p$-algebra $R$. Evaluated at finite truncation sets $S$, the relative de Rham-Witt complex $W_S\Omega_{X/R}^{\bullet}$ is a quotient of the de Rham complex $\Omega^{\bullet}_{W_S(X)/W_S(R)}$, which can be computed affine locally via explicit, but complicated relations. In this paper we prove that $W_S\Omega_{X/R}^{\bullet}$ is the torsionless quotient of the usual de Rham complex $\Omega^{\bullet}_{W_S(X)/W_S(R)}$ on the singular scheme $W_S(X)$. This result was suggested by comparison with a similar modification of the de Rham complex in the theory of singular varieties.