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Universal Deformation Rings and Quivers of Finite Representation Type

Let k be a field of arbitrary characteristic and let Q be a quiver of finite representation type. In this paper we prove that if M is an indecomposable kQ-module then the universal deformation ring of M over kQ is isomorphic to k.


Introduction
Universal deformations rings of representations of profinite Galois groups have played an important role in proving long-standing problems in number theory. The most celebrated of these is Fermat's Last Theorem (see [7]) but many other results have also been established such as the Taniyama-Shimura-Weil and Serre conjectures (see [4,5]). In general, if k is a field of arbitrary characteristic, W is a complete local commutative Noetherian ring with residue field k, G is a profinite group and V is a finite dimensional k-vector space with continuous G action, then it has been shown (see [8,13]) that if End kG (V ) ∼ = k and H 1 (G, End k (V )) is finite dimensional over k, that V has a universal deformation ring, R W (G, V ). Moreover, in [8] it was shown that R W (G, V ) is isomorphic to the inverse limit of the universal deformation rings of R W (G i , V ), where the G i range over all discrete finite quotients of G through which the action of G on V factors. As such it is important that we understand universal deformation rings for modules over finite dimensional algebras kG, where G is a finite group. Indeed deformations of modules for finite dimensional algebras have been studied by many authors in different contexts (see [10,12,14]) and in particular we use the work of Velez in [3] to prove our result.
In this paper we consider k-algebras arising from quivers of finite representation type. In Section 1 we define universal deformation rings and when they exist. In Section 2 we give a brief overview of quivers of finite representation type. Finally in Section 3 we discuss Gabriel's Theorem as it is a major result that allows us to prove our theorem below in Section 4. Theorem 1. Let Q be a quiver of finite representation type and M an indecomposable kQ-module. Then the universal deformation ring of M over kQ is isomorphic to k.

Universal deformation rings
Let k be a field of arbitrary characteristic. LetĈ be the category which has all complete local commutative Noetherian k-algebras with residue field k as objects, and continuous k-algebra homomorphisms inducing the identity map on k as morphisms.
Suppose Λ is a finite dimensional k-algebra, V is a finitely generated Λ-module and R is an object inĈ. A lift of V over R is a pair (M, φ) where M is a finitely generated R ⊗ k Λmodule which is free over R and φ is a Λ-module isomorphism φ : to be the set of all such deformations of V over R. Define the deformation functor F V :Ĉ → Sets as the covarient functor which sends a ring R inĈ to Def Λ (V, R) and a morphism α : , is bijective. In this case, R(Λ, V ) is said to be the universal deformation ring of V over R.

Quivers of Finite Representation Type
A quiver is a quadruple Q = (Q 0 , Q 1 , s, t), where Q 0 is a finite set of vertices, Q 1 is a finite set of arrows, and s, t : Q 1 → Q 0 are maps assigning to each arrow it source, resp. target. A representation M of a quiver Q consists of a family of vector spaces V i indexed by the vertices i ∈ Q 0 , together with a family of linear maps f α : . For a quiver Q and a field k we can form the category Rep k (Q) whose objects are representations of Q with the morphisms as defined above. A morphism φ : As can be expected, we wish to classify all representations of a given quiver Q up to isomorphism.
If M and N are two representations of the same quiver Q, we define their direct sum for all α ∈ Q 1 . We say that M is a trivial representation if V i = 0 for all i ∈ Q 0 . If M is isomorphic to a direct sum M 1 ⊕M 2 , where M 1 and M 2 are nontrivial representations, then M is called decomposable. Otherwise M is called indecomposable. Every representation has a unique decomposition into indecomposable representations, up to isomorphism and permutation of components. Thus the classification problem reduces to classifying the indecomposable representations. We say that a quiver is of finite representation type, or just finite type, if it has only finitely many indecomposable representations; otherwise, it is of infinite representation type. The following theorem classifying the quivers of finite representation type is due to Gabriel (see [9], [11]).

Gabriel's Theorem
The algebra of a quiver Q is the associative algebra kQ determined by the generators e i and α, where i ∈ Q 0 and α ∈ Q 1 , and the relations e 2 i = e i , e i e j = 0(i = j), e t(α) α = αe s(α) = α. Note that the category of representations of any quiver Q is equivalent to the category of left kQ-modules (see [1,Theorem III.1.5]).
Recall, if M and N are arbitrary kQ-modules we define the groups Ext i Q (M, N ) as follows: first choose a projective resolution Then take morphisms to N yielding a complex The Euler form of the quiver Q is the bilinear form , Q on R Q 0 given by for any m = (m i ) i∈Q 0 and n = (n i ) i∈Q 0 . The quadratic form associated to the Euler form is called the Tits form q Q , i.e.
q Q (n) = n, n Q = i∈Q 0 Note that the Tits form depends only on the underlying undirected graph of Q and determines the graph uniquely (see [6]). Finally a representation M of the quiver Q is called a Schur representation (also known as a brick ), if End Q (M ) ∼ = k. Clearly, any Schur representation is indecomposable, but the converse only holds if the Tits form of Q is positive definite.
Now we obtain a more precise form of Gabriel's Theorem with important consequences for our result. Note that, as a result of [6,Proposition 1.4.6], the quivers Q with positive definite Tits form are exactly the quivers which are of finite representation type. In particular we have that if Q is a quiver of finite representation type and M is an indecomposable representation of the quiver Q, i.e. an indecomposable kQ-module, then End Q (M ) ∼ = k and Ext 1 Q (M, M ) = 0.

Proof of Theorem 1
Now we can prove our result. Assume the hypotheses of Theorem 1.
Proof. By Theorem 3.1 we have that End Q (M ) ∼ = k and that Ext 1 Q (M, M ) = 0. Thus by Theorem 1.1 a universal deformation ring exists and by Theorem 1.2 we have a surjective map λ : k → R(kQ, M ). Since ker λ = 0 we obtain our isomorphism.