On almost free torus actions and Horrocks conjecture

We consider a model for cohomology groups of a space $X$ with an action of torus, representing Koszul complex of its equivariant cohomology. Studying homological properties of modules over polynomial ring we derive new estimates on homological rank (total dimension of rational cohomology) of $X$. In particular, we obtain simple proof of toral rank conjecture in the case of torus dimension $\le 4$.