Multiplicativity and nonrealizable equivariant chain complexes

Let G be a finite p-group and F field of characteristic p. We filter the cochain complex free G-space with coefficients in by powers augmentation ideal FG. show that cup product induces multiplicative structure on arising spectral sequence compute E1-page as bigraded algebra. As an application, we prove recent counterexamples Iyengar Walker to algebraic version Carlsson's conjecture can not realized topologically.