We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring $\mathbb{ F}_p[G]$ of an elementary abelian $p$-group $G$ in terms of commutative algebra. This extends results of Carlsson for $p=2$ to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over $\mathbb{ F}_p[G]$ into the derived category of $p$-DG modules over a polynomial ring.
Categories: math.AT math.AC math.RT