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We reformulate the problem of bounding the total rank of the homology ofperfect chain complexes over the group ring $\mathbb{F}_p[G]$ of an elementaryabelian $p$-group $G$ in terms of commutative algebra. This extends results ofCarlsson for $p=2$ to all primes. As an intermediate step, we construct anembedding of the derived category of perfect chain complexes over$\mathbb{F}_p[G]$ into the derived category of $p$-DG modules over a polynomialring.

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