Free $(\mathbb{Z}\mathrm{/}p{)}^n$-complexes and $p$-DG modules

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.