Functors of Artin rings

Abstract

In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation. Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u. the residue field. (In most applications A is k itself, or a ring of Witt vectors.) Let C be the category of Artin local A-algebras with residue field k. A covariant functor F from C to Sets is called pro-representable if it has the form