Let X be a smooth algebraic variety over k. We prove that any flat quasicoherent sheaf on \({\text {Ran}}(X)\) canonically acquires a \(\mathscr {D}\)-module structure. In addition, we prove that, if the geometric fiber \(X_{\overline{k}}\) is connected and admits a smooth compactification, then any line bundle on \(S \times {\text {Ran}}(X)\) is pulled back from S, for any locally Noetherian k-scheme S. Both theorems rely on a family of results which state that the (partial) limit of an n-excisive functor defined on the category of pointed finite sets is trivial.

中文翻译：

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n-Ran空间上的精确函子，规范连接和线束

令X为k的光滑代数变体。我们证明\（{\ text {Ran}}（X）\）上的任何平坦准拟相干捆都可以规范地获得\（\ mathscr {D} \）模块结构。此外，我们证明，如果几何纤维\（X _ {\ overline {k}} \）已连接并允许平滑压实，则\（S \ times {\ text {Ran}}（X ）\）从拉回小号，对于任何本地诺特ķ -scheme小号。这两个定理都依赖于一系列结果，这些结果表明，在尖的有限集类别上定义的n-精确函子的（部分）极限是微不足道的。