Equivariant Intersection Theory

In this thesis we are going to introduce the fundamental properties of equivariant Chow groups of varieties with a $G$-action, where $G$ is a linear algebraic group over any base field. This definition was firstly given by Totaro for classifying spaces and then extended to $G$-varieties by Edidin and Graham. We will make some computations (all known in the literature) of Chow rings of classifying spaces of some classical groups, starting with some relatively simple cases like the general linear group $\GL_n$, till some more involved ones as the special orthogonal group $\SO_n$ with $n$ even. This last computation will be a good excuse to develop some theory about Chow groups of oriented quadric bundles. We will then expose a general theorem, due to Edidin and Graham, which allows to identify the Chow ring of a special reductive group $G$ over an algebraically closed field with the subalgebra of $W$-invariant elements in the Chow ring of a maximal torus $T$ where $W=N(T)/T$ is the Weyl group. This theorem sinks its roots in a previous result by Vistoli that defined a ring homomorphism $\CH^*(\mathrm{BT})^W\rightarrow \CH^*(\mathrm{BG})$ working with rational coefficients.