Explicit Construction of Universal Deformation Rings

Abstract

Abstract. Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete O-algebras A with residue field k the representations of G over A that reduce to V over k are given by O-algebra homomorphisms R → A, where R is the universal deformation ring of V . We show this with an explicit construction of R. The ring R is noetherian if and only if H 1 (G, End k (V )) has finite dimension over k. IntroductionLet G be a profinite group and let k be a field. By a k-representation of G we mean a finite dimensional vector space over k with the discrete topology, equipped with a continuous k-linear action of G. If V is a k-representation of G and A is a complete local ring with residue field k, then a deformation of V in A is an isomorphism class of continuous representations of G over A that reduce to V modulo the maximal ideal of A; precise definitions are given in Section 2. We denote by Def(V, A) the set of such deformations.Let V be an absolutely irreducible k-representation of G. The object of this chapter is to give a straight-forward construction of a ring R, the universal deformation ring, which represents the functor Def(V, −). In a purely algebraic setting, without considerations of continuity, a similar construction was already given by Procesi in the seventies [9, Chap. IV, Lemma 1.7; 10]. The existence of R in the present context was deduced first by Mazur [8] with Schlessinger's criteria for pro-representability [12]. An alternative construction was given recently by Faltings (see [5] and Section 7 below).The main result of this chapter, formulated below as Theorem (2.3), is actually a little more general than Mazur's. Following Schlessinger, Mazur works only with noetherian rings, and this forces him to assume at the outset that a certain cohomology group is finite. For our argument, the noetherian condition is a hindrance, and we find it more convenient to follow Grothendieck [6] and work with not necessarily noetherian rings that are projective limits of artinian rings. This allows us to drop Mazur's cohomological condition; it reappears only at the end, as a necessary and sufficient condition for R to be noetherian.

Universal deformation rings and dihedral defect groups

“…By a result of Faltings (see [18,Prop. 7.1]), V has a universal deformation ring in case End kG (V ) = k.…”

“…Let Γ be a profinite group, and suppose V is a finite dimensional vector space over k with a continuous Γ-action. If all continuous kΓ-module endomorphisms of V are given by scalar multiplications, an argument of Faltings (see [18]) shows that V has a universal deformation ring R(Γ, V ). The topological ring R(Γ, V ) is universal with respect to deformations of V over commutative local W -algebras with residue field k which are the projective limits of their discrete Artinian quotients.…”

“…The topological ring R(Γ, V ) is universal with respect to deformations of V over commutative local W -algebras with residue field k which are the projective limits of their discrete Artinian quotients. For more information on deformation rings see [18], [27] and §2. In number theory, deformation rings are at the center of work by many authors concerning Galois representations, modular forms, elliptic curves and diophantine geometry (see e.g.…”

Universal Deformation Rings for the Symmetric Group S 4

“…In other words, R(G, V ) represents the functor F V in the sense that F V is naturally isomorphic to Hom C (R(G, V ), −). For more information on deformation rings see [13] and [19].…”

“…[12], [21,20], [9] and their references). In [13], de Smit and Lenstra showed that if Γ is an arbitrary profinite group and V is a finite dimensional vector space over k with a continuous Γ-action which has a universal deformation ring R(Γ, V ), then R(Γ, V ) is the inverse limit of the universal deformation rings R(G, V ) when G runs over all finite discrete quotients of Γ through which the Γ-action on V factors. Thus to answer questions about the ring structure of R(Γ, V ), it is natural to first consider the case when Γ = G is finite.…”