# A connection between equivariant and non-equivariant cohomology of toric variety

Let $$X$$ be a smooth projective toric variety over $$\mathbb{C}$$. It is acted by the compact torus $$T=(S^1)^n$$.

The $$T$$-equivariant cohomology $$H^*_T(X)$$ (with coefficients in a field, say) is an algebra over the ring of the $$T$$-equivariant cohomology of the point $$H^*_T(pt)$$. The ideal $$H^{>0}_T(pt)\cdot H^*_T(X)$$ is clearly two-sided.

Is it true that the quotient algebra $$H^*_T(X)/H^{>0}_T(pt)\cdot H^*_T(X)$$ is isomorphic to $$H^*(X)$$ as a graded algebra?

Yes. The keyword here is equivariantly formal. More generally if $$X$$ is a (possibly singular) projective variety over $$\mathbb{C}$$ whose ordinary cohomology $$H(X)$$ vanishes in odd degrees, and if $$X$$ admits an algebraic action of a torus $$T=(\mathbb{C}^*)^n$$ with compact torus $$K=(S^1)^n$$ then equivariant cohomology $$H_K(X)$$ is a free module over $$H_K(pt)$$ and can be obtained as extension of scalars from the ordinary cohomology, and $$H(X)\cong H_K(X)/H_K^{>0}(X)\cdot H_K(X).$$ This follows from Theorem 14.1 in this paper of Goresky-Kottwitz-MacPherson. Also Proposition 2 from this paper of Brion applies to your situation.