  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.
