N-Excisive functors, canonical connections, and line bundles on the Ran space

Abstract

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.