On the $p$-typical de Rham–Witt complex over $W(k)$

Abstract

Hesselholt and Madsen (2004) define and study the (absolute, $p$-typical) de Rham–Witt complex in mixed characteristic, where $p$ is an odd prime. They give as an example an elementary algebraic description of the de Rham–Witt complex over ${\mathbb{Z}}_{\left(p\right)}$, $W_{\cdot }{\mathrm{\Omega }}_{{\mathbb{Z}}_{\left(p\right)}}^{\bullet }$. The main goal of this paper is to construct, for $k$ a perfect ring of characteristic $p>2$, a Witt complex over $A=W\left(k\right)$ with an algebraic description which is completely analogous to Hesselholt and Madsen’s description for ${\mathbb{Z}}_{\left(p\right)}$. Our Witt complex is not isomorphic to the de Rham–Witt complex; instead we prove that, in each level, the de Rham–Witt complex over $W\left(k\right)$ surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of $p$. We deduce an explicit description of $W_n{\mathrm{\Omega }}_A^{\bullet }$ for each $n\ge 1$. We also deduce results concerning the de Rham–Witt complex over certain $p$-torsion-free perfectoid rings.