On the Chow ring of the classifying stack of PGL3,C

Equivariant intersection theory is similar to Borel’s equivariant cohomology. The common basic idea is simple. Let X be an algebraic scheme over a field k and let G be an algebraic group acting on X . Since invariant cycles are often too few to get a full-fledged intersection theory (e.g. to have a ring structure in smooth cases) we decide to enlarge this class to include invariant cycles not only on X but on X × V where V is any linear representation of G. If k = C, equivariant cohomology. can be defined along these lines and this definition agrees with the usual one given using the classifying space of G. In particular, we get a non trivial equivariant intersection theory AG = A ∗ G(pt) on pt = Speck which can be interpreted naturally as an intersection theory on the classifying stack of the group in the same way as equivariant cohomology of a point is naturally viewed as cohomology of the classifying space of the group. Equivariant intersection theory (in the sense sketched above) was first defined by Totaro in [To2] for X = Speck and then extended to general X by Edidin and Graham in [EG1]. Totaro himself ([To2])