Universal D-modules and stacks of étale germs of n-dimensional varieties

Abstract

We introduce stacks classifying étale germs of pointed varieties of dimension 𝑛. We show that quasi-coherent sheaves on these stacks are universal 𝒟-and 𝒪-modules. We state and prove a relative version of Artin's approximation theorem, and as a consequence identify our stacks with classifying stacks of automorphism groups of the 𝑛-dimensional formal disc. We introduce the notion of convergent universal modules, and study them in terms of these stacks and the representation theory of the automorphism groups.