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

Abstract

Abstract Let X be a smooth algebraic variety over k. We prove that any flat quasicoherent sheaf on

$${\text {Ran}}(X)$$

Ran ( X )

canonically acquires a

$$\mathscr {D}$$

D

-module structure. In addition, we prove that, if the geometric fiber

$$X_{\overline{k}}$$

X

k ¯

is connected and admits a smooth compactification, then any line bundle on

$$S \times {\text {Ran}}(X)$$

S × 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.