Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups

TL;DR

This paper develops Koszul resolutions for cyclic p-groups to compute Mackey functor-valued Hochschild homology in equivariant algebra, extending algebraic tools to the setting of incomplete Tambara functors.

Contribution

It introduces cyclic-p-group-equivariant Koszul resolutions for free incomplete Tambara functors, enabling the computation of Mackey functor-valued Hochschild homology.

Findings

Resolved the Burnside Mackey functor using Koszul resolutions.

Computed Hochschild homology for cyclic groups of odd prime order.

Extended algebraic techniques to equivariant settings.

Abstract

In equivariant algebra, Mackey functors replace abelian groups and incomplete Tambara functors replace commutative rings. In this context, we prove that equivariant Hochschild homology can sometimes be computed using Mackey functor-valued Tor. To compute these Tor Mackey functors for odd primes $p$, we define cyclic-$p$-group-equivariant analogues of the Koszul resolution which resolve the Burnside Mackey functor (the analogue of the integers) as a module over free incomplete Tambara functors (the analogue of polynomial rings). We apply these Koszul resolutions to compute Mackey functor-valued Hochschild homology of free incomplete Tambara functors for cyclic groups of odd prime order and for the cyclic group of order 9.