Λ is a complete Noetherian local ring with residue field k, and C is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field k. A small extension in C is a morphism Y→Z in C that is surjective with kernel a 1-dimensional vector space over k. A functor is called representable if it is of the form hX where hX(Y)=hom(X,Y) for some X, and is called pro-representable if it is of the form Y→lim hom(Xi,Y) for a filtered direct limit over i in some filtered ordered set.
If in addition the map between the tangent spaces of F and G is an isomorphism, then F is called a hull of G. Grothendieck (1960, proposition 3.1) showed that a functor from the category C of Artinian algebras to sets is pro-representable if and only if it preserves all finite limits.
By taking the projective limit of the pro-representable functor in the larger category of linearly topologized local rings, one obtains a complete linearly topologized local ring representing the functor.
One difficulty in applying Grothendieck's theorem is that it can be hard to check that a functor preserves all pullbacks.
Schessinger's theorem gives conditions for a set-valued functor F on C to be representable by a complete local Λ-algebra R with maximal ideal m such that R/mn is in C for all n. Schlessinger's theorem states that a functor from C to sets with F(k) a 1-element set is representable by a complete Noetherian local algebra if it has the following properties, and has a hull if it has the first three properties: