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

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 $W_S{\mathrm{\Omega }}_{X\mathrm{/}R}^{\bullet }$ is a quotient of the de Rham complex ${\mathrm{\Omega }}_{W_S(X)\mathrm{/}{W}_S(R)}^{\bullet }$, which can be computed affine locally via explicit, but complicated relations. In this paper we prove that $W_S{\mathrm{\Omega }}_{X\mathrm{/}R}^{\bullet }$ is the torsionless quotient of the usual de Rham complex ${\mathrm{\Omega }}_{W_S(X)\mathrm{/}{W}_S(R)}^{\bullet }$ 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.