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?**



A connection between equivariant and non-equivariant cohomology of toric variety
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](https://www.math.ias.edu/%7Egoresky/pdf/equivariant.jour.pdf) of Goresky-Kottwitz-MacPherson.
Also Proposition 2 from [this paper](https://www-fourier.ujf-grenoble.fr/%7Embrion/notesmontreal.pdf) of Brion applies to your situation.


