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In this talk we consider the ring of equivariant cohomology $H^*_{S^1_i}(\mathcal{Z}_{\mathcal{K}})$, where $S^1_i$ is the $i$th coordinate circle in $m$--dimensional torus ${T}^m$. It is known how to calculate the ring $H^*_{S^1_i}(\mathcal{Z}_{\mathcal{K}})$. We consider the following question: what are the necessary and sufficient conditions under which the equivariant cohomology ring $H^*_{S^1_i}(\mathcal{Z}_{\mathcal{K}})$ is a free module over $\mathbb{Z}[\upsilon_i]$ for $\forall \, i$. The criterion for flag and one-dimensional complexes will be given in this talk.