Universal D-modules and stacks of \'etale germs of n-dimensional varieties

We introduce stacks classifying 'etale 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.