ETD

• algebraic tori
• characteristic classes
• equivariant Chow rings

The aim of my thesis is to give an introduction to equivariant intersection theory, as developed by Burt Totaro, Dan Edidin and William Graham. I will give the basic theoretical notions, expose some known example and finally try to do some calculation for non-split tori, which do not seem to have been treated in the literature.
If G is an algebraic group over a field k, denote by A(G) the equivariant Chow ring of G acting on Spec(k); this is what Totaro calls the Chow ring of the classifying space of G.
I have studied the basis of intersection theory in W. Fulton's book; then I have read some of the foundational papers in equivariant intersection theory, such as The Chow ring of a classifying space, by B. Totaro, Equivariant intersection theory, by D. Edidin and W. Graham, and On the Chow ring of classifying spaces for classical groups. Thus I have learned many standard technique for computing
Chow rings of classifying spaces. The first few chapters of my thesis will consist of an exposition of this theory, and then of a review of the calculation of A(G) in some known case, such as that of GLn;On.
Then I will try to compute A(T) for some simple algebraic tori T over k. The case in which T is totally split is well understood; I have been looking at permutation tori, i.e., tori that come by Weil restriction from a finite separable field extension k'/k. In this case the Chow ring A^*_T is equal to the subring of invariants of the character ring X(T) by the action of the Galois group Gal(k'/k).