The moduli stack of $G$-bundles

Abstract

In this paper, we give an expository account of the geometric properties of the moduli stack of $G$-bundles. For $G$ an algebraic group over a base field and $X\to S$ a flat, finitely presented, projective morphism of schemes, we give a complete proof that the moduli stack $Bu{n}_G$ is an algebraic stack locally of finite presentation over $S$ with schematic, affine diagonal. In the process, we prove some properties of $BG$ and Hom stacks. We then define a level structure on $Bu{n}_G$ to provide alternative presentations of quasi-compact open substacks. Finally, we prove that $Bu{n}_G$ is smooth over $S$ if $G$ is smooth and $X\to S$ is a relative curve.

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