Hochschild-Witt complex

Abstract

In a previous paper, 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 WCH⁎(A) for any associative unital k-algebra A, with homology groups WHH⁎(A). We prove that the group WHH0(A) coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if A is commutative, finitely generated, and smooth, the groups WHHi(A) are naturally identified with the terms WΩAi of the de Rham-Witt complex of the spectrum of A.