Spectrum of equivariant cohomology as a fixed point scheme

An action of a complex reductive group $\mathrm{G}$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm{G}$ act with finitely many fixed points. Then the complex $\mathrm{G}$-equivariant cohomology ring of $X$ is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.