Noncommutative relative de Rham--Witt complex via the norm

Abstract

In [Ill79], Illusie constructed de Rham-Witt complex of smooth $\mathbb F_p$-algebras R, which computes the crystalline cohomology of R, a $\mathbb Z_p$-lift of the de Rham cohomology of R. There are two different extensions of de Rham-Witt complex: a relative version discovered by Langer-Zink, and a noncommutative version, called Hochschild-Witt homology, constructed by Kaledin. The key to Kaledin's construction is his polynomial Witt vectors. In this article, we introduce a common extension of both: relative Hochschild-Witt homology. It is simply defined to be topological Hochschild homology relative to the Tambara functor $W(\mathbb F_p)$. Adopting Hesselholt's proof of his HKR theorem, we deduce an HKR theorem for relative Hochschild-Witt homology, which relates its homology groups to relative de Rham-Witt complex. We also identify Kaledin's polynomial Witt vectors as the relative Hill-Hopkins-Ravenel norm, which allows us to identify our Hochschild-Witt homology relative to $\mathbb F_p$ with Kaledin's Hochschild-Witt homology. As a consequence, we deduce a comparison between Hochschild-Witt homology and topological restriction homology, fulfilling a missing part of [Kal19].