  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.
