  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.
