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

TL;DR

The paper proves that the Beilinson--Drinfeld affine Grassmannian over a smooth curve is a colimit of reduced ind-schemes, simplifying the understanding of its geometric structure and factorization properties.

Contribution

It establishes that $ ext{Gr}_{G, ext{Ran}(X)}$ is the colimit of reduced ind-schemes and generalizes the notion of scheme reduction to presheaves, enhancing the understanding of their geometric behavior.

Findings

$ ext{Gr}_{G, ext{Ran}(X)}$ is a colimit of reduced ind-schemes.

Every map from an affine scheme factors through a reduced quasi-projective scheme.

Generalized the reduction of schemes to presheaves, applicable to pseudo-ind-schemes.

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.