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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?

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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.

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