${\mathrm{Gr}}_{G,\mathrm{Ran}(X)}$ is reduced

Abstract

Let $k$ be a field of characteristic zero. Fix a smooth algebraic curve $X$ and a split reductive group $G$ over $k$. We show that the Beilinson--Drinfeld affine Grassmannian ${\mathrm{Gr}}_{G,\mathrm{Ran}(X)}$ is the presheaf colimit of the reduced ind-schemes $({\mathrm{Gr}}_{G,X^I}{)}^{\mathrm{red}}$ for finite sets $I$. This implies that every map from an affine $k$-scheme to ${\mathrm{Gr}}_{G,\mathrm{Ran}(X)}$ factors through a reduced quasi-projective $k$-scheme. In the course of the proof, we generalize the notion of 'reduction of a scheme' to apply to any presheaf, and we show that this notion is well-behaved on any pseudo-ind-scheme which admits a colimit presentation whose indexing category satisfies the amalgamation property.

27 pages, LaTeX; minor improvements to exposition and terminology