Homotopy Mackey functors of equivariant algebraic $K$-theory

TL;DR

This paper computes the homotopy Mackey functors of equivariant algebraic K-theory using an algebraic approach, making the structure more explicit and accessible, especially for trivial group actions.

Contribution

It provides an algebraic construction of the homotopy Mackey functors for equivariant algebraic K-theory, simplifying their computation and interpretation.

Findings

Explicit algebraic description of Mackey functors from algebraic K-groups.

Recovery of classical Mackey functors for trivial group actions.

Development of new examples of Mackey functors in algebraic K-theory.

Abstract

Given a finite group $G$ acting on a ring $R$, Merling constructed an equivariant algebraic $K$-theory $G$-spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide a computation of the homotopy Mackey functors of equivariant algebraic $K$-theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the $n$th algebraic $K$-groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along module categories of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic $K$-theory of group rings. We develop many families of examples of Mackey functors, both new and old, including $K$-theory of endomorphism rings, the $K$-theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings.