Homotopy Mackey functors of equivariant 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 nth 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. Contents 1. Introduction 1 2. Preliminaries 4 3. Mackey functors on algebraic K-theory of twisted group rings 13 4. Comparison with equivariant algebraic K-theory 23 5. Families of Mackey functors 32 Appendix A. Bimodule structures 35 References 38

Genuine vs. naïve symmetric monoidal G-categories

“…In our situation there are natural candidates for the restrictions and transfers along H ⊂ K: namely, they ought to correspond under the equivalence (5.1) to restriction or extension of scalars along RH → RK, respectively. However, while it is plausible that one would be able to prove this by a combination of the methods from [Len22] and [Bra21], such a comparison is far from being trivial, and we will not pursue this any further here.…”