Topology of Algebraic Varieties

We will first quickly recall basic facts on the tree of SL_2 over a field K with a discrete valuation v, following Serre's book. We will then generalize the geometric interpretation given in that book for curves to a higher dimensional situation where K = F(T_1,..,T_n) is the function field of P^n over F and v is the valuation at infinity, in terms of reflexive coherent sheaves on P^n. We will describe applications of this, for instance to give new results on the homology of SL_2(F[T_1,..,T_n]) . We will also construct new morphisms between these kind of trees, like restriction morphisms corresponding to a linear embedding P^{n-1} --> P^n (transverse to the infinity). We will finally use this geometric interpretation to study the local structure of these trees around a vertex, and sketch how all this can be used to prove the weak homotopy invariance property for the homology of SL_2 over field F in which any element is a square, a crucial step toward the proof of Friedlander-Milnor conjecture for SL_2 . All this can be generalized to higher dimensional split groups, affine buildings and suitable fields F.