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

TL;DR

This paper extends bounds on the homology of perfect chain complexes over elementary abelian p-groups to all primes by connecting group ring complexes with p-DG modules via commutative algebra.

Contribution

It introduces a new approach linking perfect chain complexes over group rings to p-DG modules, generalizing previous results from p=2 to all primes.

Findings

Bound on the total rank of homology extended to all primes.

Constructed an embedding of derived categories into p-DG modules.

Unified framework for analyzing chain complexes over elementary abelian p-groups.

Abstract

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.