  Let $G$ be a finite $p$-group and $\mathbb{F}$ a field of characteristic $p$.
We filter the cochain complex of a free $G$-space with coefficients in
$\mathbb{F}$ by powers of the augmentation ideal of $\mathbb{F} G$. We show
that the cup product induces a multiplicative structure on the arising spectral
sequence and compute the $E_1$-page as a bigraded algebra. As an application,
we prove that recent counterexamples of Iyengar and Walker to an algebraic
version of Carlsson's conjecture can not be realized topologically.
