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

Abstract

We introduce stacks classifying étale germs of pointed n-dimensional varieties. 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.

61 pages. Version 1 had a gap: Artin's approximation theorem was misstated and the incorrect version was used. This gap has been fixed, using new material in sections 2 and 5. Section 8 has been added, to treat the dg-categorical version of the results. The paper has been restructured and the introduction has been expanded. Version 3: minor changes