Hochschild-Witt complex

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field $k$ of positive characteristic $p$ to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex $WC{H}_{\ast }(A)$ for any associative unital $k$-algebra $A$, with homology groups $WH{H}_{\ast }(A)$. We prove that the group $WH{H}_0(A)$ coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if $A$ is commutative, finitely generated, and smooth, the groups $WH{H}_i(A)$ are naturally identified with the terms $W{\mathrm{\Omega }}_A^i$ of the de Rham-Witt complex of the spectrum of $A$.