Pathological Computations of Mackey Functor-valued Tor over Cyclic Groups

TL;DR

This paper computes Mackey functor-valued Tor over specific Tambara functors, revealing nonvanishing higher-degree Tor groups and exploring analogues of Koszul complexes in a 2-primary setting.

Contribution

It generalizes Tor computations to Mackey functor-valued contexts and introduces new examples with nonvanishing higher-degree Tor groups.

Findings

Tor groups can be nonvanishing in almost every degree.

Extension of Koszul complexes to 2-primary analogues.

Generalization of classical Tor computations to Mackey functor settings.

Abstract

We compute Mackey functor-valued Tor over certain free incomplete Tambara functors, generalizing the computation of Tor over a polynomial ring on one generator. In contrast with the classical situation where the resulting Tor groups vanish above degree one, we present examples where Tor is nonvanishing in almost every degree. We also discuss a 2-primary analogue of the odd-primary Koszul complexes defined in our other work \cite{MQS24a}.