论文标题
$ \ mathrm {gr} _ {g，\ mathrm {ran}（x）} $减少
$\mathrm{Gr}_{G, \mathrm{Ran}(X)}$ is reduced
论文作者
论文摘要
令$ k $为特征零字段。修复平滑的代数曲线$ x $和一个拆分还原组$ g $以上$ k $。我们表明，贝林森 - 德林菲尔德·格拉斯曼尼亚$ \ mathrm {gr} _ {g，\ mathrm {ran}（x）（x）} $是减少的ind-schemes $（\ m iandrm {gr} _ {g，x，x^i} $ fin in in ind-schemes $（ $ i $。这意味着每张从仿射$ k $ -scheme到$ \ mathrm {gr} _ {g，\ mathrm {ran}（x）（x）} $因素的每张地图。在证明过程中，我们将“减少计划的减少”的概念概述适用于任何前eaf，我们表明，这种概念在任何伪Ind-scheme上都表现得很好，后者承认其索引类别满足合并特性的索引类别。
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.
