  We use the multiplicative structure of the Koszul resolution to give short
and simple proofs of some known estimates for the total dimension of the
cohomology of spaces which admit free torus actions and analogous results for
filtered differential modules over polynomial rings. We also point out the
possibility of improving these results in the presence of a multiplicative
structure on the so-called minimal Hirsch-Brown model for the equivariant
cohomology of the space.
