The Generically Trivial Case ( Lecture 11 )

Throughout this lecture, we let k denote an algebraically closed field, a prime number which is invertible in k. Let X be an algebraic curve over k, G a smooth affine group scheme over X whose generic fiber is semisimple and simply connected. Our goal, over the next several lectures, is to prove the following result: Theorem 1. Let R be a finitely generated k-algebra, let P be a G-bundle over the relative curve X R = Spec R × Spec k X, and let Sect(P) be the prestack parametrizing rational sections of P. Then the projection map Sect(P) → Spec R induces an isomorphism on Z-homology. There are several (equivalent) versions of this statement, depending on our definition of " rational section " of P. More precisely, by varying the definitions of the previous lecture, we can produce a diagram of prestacks Sect(P) / / Sect(P) + Sect u (P) / / Sect u (P) + , where each of the morphisms is a universal homological equivalence (by the results of the last lecture). We are therefore free to pick whichever definition of " rational section " is best suited to our purposes at the moment. In this lecture, it will be convenient to work with the prestack Sect u (P) + , which we can describe precisely as follows: • An object of Sect u (P) + is a triple (A, S, γ) where A is a finitely generated R-algebra, S is a finite subset of X(A), and γ fits into a commutative diagram