Quotients of Functors of Artin Rings

One of the fundamental problems in the study of moduli spaces is to give an intrinsic characterisation of representable functors of schemes, or of functors that are quotients of representable ones of some sort. Such questions are in general hard, leading naturally to geometry of algebraic stacks and spaces (see [1, 3]). On the other hand, in infinitesimal deformation theory a classical criterion due to Schlessinger [4] does describe the pro-representable functors and, more generally, functors that have a hull. Our result is that in this setting the question of describing group quotients also has a simple answer. In other words, for functors of Artin rings that have a hull, those that are quotients of pro-representable ones by a constant group action can be described intrinsically. To set up the notation, let Λ be a complete Noetherian local ring with residue field k, and fix an isomorphism Λ/mΛ ∼= k. We write ArtΛ for the category of Artinian local Λ-algebras A given together with an augmentation A/mA ∼= k. Morphisms (denoted HomΛ(A,B)) are local homomorphisms of Λ-algebras that commute with the augmentations. Every complete Noetherian local Λ-algebra F with an augmentation (not necessarily Artinian) gives rise to a covariant functor