The Witt vectors for Green functors

Abstract

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology THH Cn (−), and under flatness conditions it describes the E 2 term of the Künneth spectral sequence for relative THH. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.

A shadow framework for equivariant Hochschild homologies

“…In particular, in [3], Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell define a theory of C n -twisted topological Hochschild homology for a C n -equivariant ring spectrum, THH Cn (R). This theory has an algebraic analogue, the twisted Hochschild homology of Green functors, developed in [9]. In this section we set up a framework for Hochschild theories twisted by an automorphism, as a particular case of the constructions in the previous section.…”

“…In recent years, equivariant generalizations of Hochschild homology and topological Hochschild homology have been developed. In [3] the authors define C n -twisted topological Hochschild homology for a C n -equivariant ring spectrum R, denoted THH Cn (R), where C n is the cyclic group of order n. Twisted topological Hochschild homology has an algebraic analogue as well, twisted Hochschild homology for Green functors, HH Cn k , as defined in [9]. In the classical setting the Hochschild homology of a ring R and its topological analogue, THH(R) are related by a map π k THH(R) → HH k (R).…”

“…where π Cn * denotes the homotopy Mackey functors for the group C n . In [9], the authors construct this linearization map and prove it is an isomorphism in degree 0.…”

“…This perspective on linearization maps also allows us to define a new linearization map, from topological restriction homology (TR) to algebraic restriction homology, tr. This algebraic version of TR, tr, was defined in [9] using Hochschild homology for Green functors. We prove that tr is a bicategorical shadow, and conclude that it is Morita invariant.…”

“…The Dennis trace map from algebraic K-theory to Hochschild homology factors through THH. It was shown in [9] that the Dennis trace map also factors through Hochschild homology for Green functors, and through algebraic tr.…”