LAFFORGUE “BACKGROUND SEMINAR” PART 3 - UNIFORMIZATION OF BunG

For us, G-bundles are taken in the fppf topology (or, equivalently, the étale topology). Remember that BunG depends on the choice of X, even though the notation suppresses this! Last time, Francois proved that BunGLn is an algebraic stack: • It satisfies descent in the fppf (equivalently, étale) topology; i.e., it’s a stack; • Its diagonal morphism is representable by an algebraic space (implying that any morphism from an algebraic space to BunGLn is representable); • It possesses a smooth cover by a scheme. Actually, we proved a bit more: BunGLn is locally of finite presentation over k and its diagonal is of finite presentation and is represented by an affine scheme. The moduli stack is not, however, quasicompact, as it has infinitely many components, indexed by degree. We can use this result to prove the same thing for arbitrary G: