ИСТИНА

In the article [2] T.E.Panov and M.Masuda considered the problem on cohomology rigidity of the family of moment-angle manifolds. For 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 ${T}^m$, we have a ring isomorphism $$ H^*_{S^1_i}(\mathcal{Z}_\mathcal{K}) \cong Tor_{\mathbb{Z}[ \upsilon_1, \dots, \upsilon_m]}(\mathbb{Z}[ \mathcal{K}], \mathbb{Z}[\upsilon_i]) \cong H(\Lambda[u_1, \dots, \hat{u_i},\dots, u_m]\otimes\mathbb{Z}[\mathcal{K}], d), $$ where in the latter algebra $u_i$ is dropped and the differential algebra is given by $du_j = \upsilon_j, d\upsilon_j = 0$, and $\mathbb{Z}[ \mathcal{K}]$ is the face ring of $\mathcal{K}$. The following question was raised: 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 $\forall \, i$. In this direction we were able to derive the next criterion for flag complexes: \begin{theorem*} The equivariant cohomology ring $H^*_{S^1_i}(\mathcal{Z}_{\mathcal{K}})$ is a free $\mathbb{Z}[\upsilon_i]$ - module for any $i$ if and only if the simplicial complex is $$\mathcal{K} = \partial\Delta^{k_1}*\dots*\partial\Delta^{k_p}*\Delta^l, l \geqslant -1, k_i \geqslant 0.$$ \end{theorem*} The same criterion holds also for one-dimensional simplicial complexes (graphs).