Carlsson conjectured that the sum of $mod-p$ Betti numbers of a finite, free $(Z/p)^n-CW$ complex is at least $2^n$. For $p=2$, he connected chain complexes with a free $(Z/p)^n$-action to DG modules over a polynomial ring in order to establish lower bounds using commutative algebra and algebraic geometry. In this talk, I will report on joint work with Jeremiah Heller on how to extend this connection to commutative algebra to all primes, and on joint work with Henrik Rüping about non-realizability of $(Z/p)^n$-equivariant chain complexes.