Versal deformation rings of modules over Brauer tree algebras

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 05 Versal deformation rings of modules over Brauer tree algebras Daniel Joseph Wackwitz University of Iowa Copyright 05 Daniel Joseph Wackwitz This dissertation is available at Iowa Research Online: Recommended Citation Wackwitz, Daniel Joseph "Versal deformation rings of modules over Brauer tree algebras" PhD (Doctor of Philosophy) thesis, University of Iowa, 05 Follow this and additional works at: Part of the Mathematics Commons

2 VERSAL DEFORMATION RINGS OF MODULES OVER BRAUER TREE ALGEBRAS by Daniel Joseph Wackwitz A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa August 05 Thesis Supervisor: Professor Frauke Bleher

3 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PHD THESIS This is to certify that the PhD thesis of Daniel Joseph Wackwitz has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Mathematics at the August 05 graduation Thesis Committee: Frauke Bleher, Thesis Supervisor Victor Camillo Philip Kutzko Miodrag Iovanov Ryan Kinser

4 To my beautiful wife and daughter all of this and all I am is for you ii

5 ACKNOWLEDGEMENTS I would first like to acknowledge and give thanks to my advisor, Doctor Frauke Bleher, for her support, advice and hard work throughout my graduate school career It has always been greatly appreciated and has been a driving force to get me to where I am today I would like to thank my wife for her constant support over the years we have had together Whenever I have felt weak or have lost faith in myself, she has been my rock I am infinitely better off having her in my life and am incredibly thankful for all she does I would like to thank my parents for their unfailing support, not just during graduate school, but throughout the entirety of my life My Dad has always pushed me to do better and has inspired me by his incredible example He has always shown me that something is not worth doing unless it is done with the utmost effort and to the best of my ability I have become the man I am today by trying to be more like him My Mom has always given me the love and support I have needed in every situation Her unwavering belief in me and her constant words of encouragement have given me the strength to make it through the most challenging situations I would also like to thank by three siblings, Kim, Brian and Jeff I have been incredibly lucky to have such great role models all my life Their examples have helped make me who I am today and have inspired me to strive for greatness Finally, I would like to thank all of my friends I have made in both underiii

6 graduate and graduate school There have been some tough times, but we have made it through them together Whether it was an unusually tough homework problem, studying for an upcoming exam, or working through some difficult research, I am grateful that I had such a great support system at school to help me through any situation iv

7 ABSTRACT This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal deformation rings of modules for these algebras The main motivation for studying Brauer tree algebras is that they generalize p-modular blocks of group rings with cyclic defect groups We consider the case when Λ is a Brauer tree algebra over an algebraically closed field k and determine the structure of the versal deformation ring R(Λ, V ) of every indecomposable Λ-module V when the Brauer tree is a star whose exceptional vertex is at the center The ring R(Λ, V ) is a complete local commutative Noetherian k-algebra with residue field k, which can be expressed as a quotient ring of a power series algebra over k in finitely many commuting variables The defining property of R(Λ, V ) is that the isomorphism class of every lift of V over a complete local commutative Noetherian k-algebra R with residue field k arises from a local ring homomorphism α : R(Λ, V ) R and that α is unique if R is the ring of dual numbers k[t]/(t ) In the case where Λ is a star Brauer tree algebra and V is an indecomposable Λ-module such that the k-dimension of Ext Λ(V, V ) is equal to r, we explicitly determine generators of an ideal J of k[[t,, t r ]] such that R(Λ, V ) = k[[t,, t r ]]/J v

8 PUBLIC ABSTRACT This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal and universal deformation rings of representations A Brauer tree algebra Λ over an algebraically closed field k can be described by a directed graph together with relations A representation V of Λ is given by assigning a finite dimensional vector space to each vertex and a linear map to each arrow of the directed graph such that the relations of the algebra are satisfied The first step is to lift such a representation V to various rings R that are quotient rings of power series rings over k in finitely many commuting variables The goal is then to use these lifts to find what is called the versal deformation ring R(Λ, V ) of V This ring has the property that it can be used to describe the isomorphism class of every lift of V over any ring R as above In this thesis, we determine the versal deformation ring of every indecomposable representation of a Brauer tree algebra in the case where the Brauer tree is a star whose exceptional vertex is at the center vi

9 TABLE OF CONTENTS LIST OF FIGURES ix CHAPTER INTRODUCTION Motivation Overview DEFINITIONS AND BACKGROUND Brauer Tree Algebras Symmetric Algebras 6 Quivers and Path Algebras 7 4 Special Biserial Algebras 9 5 Quivers and Relations for Brauer Tree Algebras 4 6 Projective Covers and the Stable Module Category 5 7 Morita and Stable Equivalences 7 8 Almost Split Sequences and Auslander-Reiten Quivers 8 9 Homomorphisms and Almost Split Sequences for String Modules of Special Biserial Algebras 0 The Stable Auslander-Reiten Quiver for Brauer Tree Algebras 4 Versal Deformation Rings 9 VERSAL DEFORMATION RINGS OF MODULES FOR BRAUER TREE ALGEBRAS 40 Dimensions of Ext Λ (M, M) and End Λ(M) for an indecomposable Λ-module M 40 Reduction to Star Brauer Tree Algebras 44 Cases to be Considered 44 4 Results 46 4 A PARTICULAR CASE 50 4 Lifts of V,0 over the Ring of Dual Numbers 50 4 Lifting V,0 over Larger Rings 5 4 Construction of a Lift of V,0 over k[[t, t ]]/(t, t, t t ) 5 4 Construction of a Lift of V,0 over k[[t, t ]]/(t, t, t t ) 58 4 Construction of a Lift of V,0 over k[[t, t ]]/(t +t t, t, t t ) 6 vii

10 44 Construction of a Lift to k[[t, t ]]/(t t + t, t + t t ) 66 4 The Versal Deformation Ring of V,0 7 5 THE GENERAL CASE 75 5 The Λ-module structure of V n,i 77 5 Lifts of V n,i over the Ring of Dual Numbers 80 5 Construction of E,, E n 80 5 Linear independence of E,, E n in Ext Λ (V n,i, V n,i ) 84 5 The Versal Deformation Ring of V n,i for i The Versal Deformation Ring of V n,i for i = 0 99 APPENDIX 07 REFERENCES 09 viii

11 LIST OF FIGURES Figure Stable Auslander-Reiten quiver for Λ in Example 8 Stable Auslander-Reiten quiver for Λ in Example 4 0 ix

12 CHAPTER INTRODUCTION Motivation Let k be an algebraically closed field, and let Λ be a finite dimensional k- algebra In this thesis, we apply methods from representation theory to study versal deformation rings of Λ-modules These versal deformation rings are complete local commutative Noetherian k-algebras with residue field k, which can be expressed as quotient rings of power series algebras over k in finitely many commuting variables We consider the case when Λ is a so-called Brauer tree algebra and determine the structure of all versal deformation rings of indecomposable Λ-modules when the Brauer tree is a star whose exceptional vertex is at the center Note that all indecomposable modules for a star Brauer tree algebra are uniserial, which means that these algebras are in particular Nakayama algebras Since there is always a stable equivalence of Morita type between an arbitrary Brauer tree algebra and a star Brauer tree algebra, this is the most important case to study in this context The main motivation for determining versal deformation rings of modules for Brauer tree algebras is that these algebras generalize p-modular blocks of group rings with cyclic defect groups The study of versal and universal deformation rings of these cyclic blocks was begun in [], where the discussion was restricted to modules whose stable endomorphisms are all given by scalar multiplications In this thesis, we consider all indecomposable modules, which means that we determine the versal

13 mod p deformation rings of all indecomposable modules belonging to cyclic p-modular blocks Overview In Chapter, we give some necessary definitions and results that will be used in the remainder of the thesis We define Brauer tree algebras, symmetric algebras, quivers and path algebras, special biserial algebras, stable module categories, Morita and stable equivalences, Auslander-Reiten quivers, and versal deformation rings In Chapter, we show how to reduce from the case of an arbitrary Brauer tree algebra to the case of a Brauer tree algebra Λ whose Brauer tree is a star with exceptional vertex at its center We also introduce some notation and state our main result, Theorem In Chapter 4, we consider a particular case, namely when the Brauer tree is a star with one edge and multiplicity We consider uniserial modules of length in this case We explicitly construct lifts of these modules over rings of larger and larger k-dimension until we arrive at the versal deformation rings In Chapter 5, we look at the case of a general star Brauer tree algebra Λ and determine the versal deformation ring of every indecomposable Λ-module whose stable endomorphism ring is not isomorphic to k In particular, we prove our main result, Theorem In the appendix, we give an explicit description of some of the ideals considered in Chapter

14 CHAPTER DEFINITIONS AND BACKGROUND Throughout this chapter, k is an algebraically closed field, and Λ is a finite dimensional k-algebra All modules are assumed to be finitely generated left modules The category of all finitely generated Λ-modules is denoted by Λ-mod The focus of this thesis is on Brauer tree algebras We first introduce these algebras and illustrate the definitions with examples We then discuss the main properties of Brauer tree algebras that are relevant for this thesis Brauer Tree Algebras Definition A Brauer tree T is a finite acyclic undirected connected graph together with a counterclockwise ordering of the edges emanating from each vertex, and a single exceptional vertex assigned a positive integer value m called the multiplicity Definition A finite dimensional k-algebra Λ is said to be a Brauer tree algebra for a given Brauer tree T (Λ) if there is a bijection between the edges j of the tree and the isomorphism classes of the simple Λ-modules S j so that the projective cover P j of S j can be described in the following way We have that P j /rad(p j ) = top(p j ) = soc(p j ) = S j, and rad(p j )/soc(p j ) is a direct sum of two (possibly zero) uniserial modules U j and V j corresponding to the vertices u = u j and v = v j adjacent to the edge j If the edges around u are counterclockwise cyclically ordered j, j,, j r, j and the multiplicity of u is m u (which is one except at the exceptional vertex), then U j has composition factors from top to bottom: S j, S j,, S jr, S j, S j,, S jr, S j,, S jr,

15 4 so that each S ji appears exactly m u times and S j appears m u times V j is defined similarly with the edges around the vertex v Example Consider the Brauer tree algebra Λ with the Brauer tree T (Λ) below, which has its edges labeled by the isomorphism classes of the simple Λ-modules, represented by the integers through 4: 4 m= Using Definition, the isomorphism classes of the projective indecomposable Λ-modules can be pictured as follows: P : P : P : 4 P 4 : Example 4 Consider the Brauer tree algebra Λ for the star Brauer tree with e edges, multiplicity m and exceptional vertex at the center, again with the edges labeled by the isomorphism classes of the simple Λ-modules: e For this algebra, every projective indecomposable Λ-module is uniserial of length me + For example, if e = and m =, the isomorphism classes of the projective indecomposable Λ-modules can be pictured as follows:

16 5 P : P : P : The following definition is from [4] Definition 5 (i) A clockwise walk W around the Brauer tree T (Λ) is a finite sequence of edges X,, X n and vertices v, v n+, written as W = (v, X, v,, v n, X n, v n+ ), where n 0, such that (a) v i v i+ are the end points of X i, (b) X i+ is the edge that is the next clockwise edge to X i around the vertex v i+ If v i+ is a leaf vertex, meaning X i is the only edge adjacent to v i+, then X i+ = X i (ii) The walk W is called a complete walk of multiplicity ω if n = eω Note that if W is a complete walk of multiplicity ω, the walk goes around T (Λ) exactly ω times, and each edge of T (Λ) appears exactly ω times in W Example 6 Consider the Brauer tree T (Λ) from Example, this time additionally labeling the vertices by v through v 5 :

17 6 v 4 v v 4 v 5 v One possible clockwise walk around T (Λ) would be W = (v,, v 4, 4, v 5, 4, v 4 ) The clockwise walk W = (v,, v,, v 4, 4, v 5, 4, v 4,, v,, v,, v,, v ) is a complete walk of multiplicity Proposition 7 Let Λ be a Brauer tree algebra Then: (i) Λ is a symmetric k-algebra, (ii) Λ is a special biserial algebra, (iii) Λ has finite representation type, meaning there are only finitely many isomorphism classes of indecomposable Λ-modules Proof These properties follow from [, Section 48] and [] We now recall the definitions of symmetric and special biserial algebras We moreover describe the isomorphism classes of all indecomposable Λ-modules in the case where Λ is a special biserial algebra of finite representation type Many of our definitions and results are taken from [5, Chapter ] Symmetric Algebras Let Λ be a finite dimensional k-algebra Definition 8 The algebra Λ is said to be a symmetric algebra if there is a linear map λ : Λ k such that Ker(λ) has no non-zero left or right ideal, and for all

18 7 a, b Λ, λ(ab) = λ(ba) We say Λ is self-injective if it is injective as a module over itself, that is if Λ Λ is an injective Λ-module Proposition 9 (i) If Λ is a symmetric algebra over k then Λ is self-injective (ii) Suppose Λ is self-injective and let M be a finitely generated Λ-module The following are equivalent: (a) M is projective (b) M is injective (c) Hom k (M, k) is projective (d) Hom k (M, k) is injective Proof See [, Proposition 6] Quivers and Path Algebras Definition 0 A quiver is a directed graph Q = (Q 0, Q, s, e) consisting of a set of vertices Q 0, a set of arrows Q, and two maps s, e : Q Q 0 where s associates to each arrow α Q the vertex s(α) at which α begins, and e associates to each arrow α Q the vertex e(α) at which α ends A quiver Q is said to be finite if Q 0 and Q are both finite sets Definition Let Q = (Q 0, Q, s, e) be a quiver (i) A path of length l in Q from vertex a to vertex b is a sequence (α l,, α, α ) where α j Q for all j l, and where s(α ) = a, e(α l ) = b, and e(α j ) =

19 8 s(α j+ ) for all j l We denote this path by α l α α We also associate a path of length l = 0 to each vertex i Q 0, which we call the trivial path at i and which we denote by e i (ii) The path algebra kq of Q is defined to be the k-vector space whose k-basis is the set of all paths in Q The product of two paths α l α α and β k β β in kq is α l α α β k β β if e(β k ) = s(α ) and 0 otherwise This defines a k-algebra structure on kq (iii) Let J be the ideal of the path algebra kq which is generated by all paths of length An ideal I of kq is said to be admissible if there exists some s such that J s I J Proposition Let Q be a quiver and let I be an admissible ideal of kq Then the image J of J in kq/i is equal to rad(kq/i) Proof See [, Proposition III6] Definition Let Q be a finite quiver A representation V = (V i, φ α ) of Q over k is defined as follows: (a) To each vertex i Q 0 a k-vector space V i is associated (b) To each arrow α : i j in Q a k-linear map φ α : V i V j is associated The representation V is said to be finite dimensional if each vector space V i is finite dimensional Let V = (V i, φ α ) and V = (V i, φ α) be two representations of Q over k A morphism

20 9 of representations f : V V is a family of k-linear maps f i : V i V i for each i Q 0 such that φ α f a = f b φ α for each arrow α : a b in Q Let Rep k (Q) be the category of finite dimensional representations of Q over k Theorem 4 The categories Rep k (Q) and kq-mod are equivalent categories Proof See [, Theorem III5] Definition 5 Recall that k is an algebraically closed field The k-algebra Λ is said to be a basic algebra if all simple Λ-modules are -dimensional over k In general, the basic algebra Λ 0 of Λ is defined as follows Write Λ = ɛ ij where the ɛ ij are orthogonal primitive idempotents, such that Λɛ ij = Λɛkl if and only if i = k Define ɛ = ɛ i and Λ 0 = ɛλɛ (see also Theorem 5) Theorem 6 (Gabriel) Suppose Λ is basic Then there exists a unique quiver Q and an admissible ideal I such that Λ = kq/i Proof See [, Corollary III0] 4 Special Biserial Algebras Definition 7 The algebra Λ is said to be a special biserial algebra if its basic algebra has the form kq/i, for a unique quiver Q and an admissible ideal I, satisfying the following conditions: (i) At most two arrows start at any vertex i of Q, and at most two arrows end at any vertex i of Q

21 0 (ii) Given an arrow β there is at most one arrow γ such that s(β) = e(γ) and βγ I, and given an arrow ɛ there is at most one arrow δ such that s(δ) = e(ɛ) and δɛ I Definition 8 The algebra Λ is called a string algebra if Λ is special biserial and its basic algebra has the form kq/i where I is generated by paths Definition 9 Let Λ = kq/i be a basic string algebra Given an arrow β of Q, let β denote a formal inverse of β, and define s(β ) = e(β), e(β ) = s(β), and (β ) = β Define an alphabet for Λ by taking as letters the arrows of Q and their formal inverses (i) A word of length n is defined to be a sequence w = w w w n where each w i is either an arrow or a formal inverse, and where s(w i ) = e(w i+ ) for i n We define s(w) = s(w n ), e(w) = e(w ), and w = wn w w For each vertex i of Q, we define an empty word of length 0, denoted by e i, where s(e i ) = e(e i ) = i and e i = e i (ii) We define an equivalence relation on the set of all words by w w if w = w or w = w We define an equivalence relation r on the set of all words w of length at least which satisfy s(w) = e(w) by w r w if either w or w is obtained from w by a cyclic rotation (iii) Let S be a complete set of representative words w = w w w n under the relation such that either n = 0, ie w is an empty word, or w i w i+ for

22 i n, and no subpath of w or w belongs to I The elements of S are called strings (iv) Let B be a complete set of representative words w = w w w n with n and s(w) = e(w) under the relation r such that w i w i+ for i n, w n w, w is not a power of a smaller word, and no subword of w m belongs to I for all m The elements of B are called bands Definition 0 Let Λ = kq/i be a string algebra (i) Let w = w w w n or w = e i be a string of length n 0 Let Q w be the quiver with underlying graph w w n wn where the edge labeled w i points to the left if w i is an arrow and to the right otherwise Define a functor G w : k-mod Λ-mod as follows: If V is a k-vector space, define G w (V ) to be the Λ-module which is defined by the representation of Q w, which assigns to each vertex of Q w the vector space V and to each arrow in Q w the identity linear transformation (ii) Let w = w w w n be a band, and assume, without loss of generality, that w

23 is an arrow Let Q w be the circular quiver w n w w n w where the edge labeled w i points counterclockwise if w i is an arrow and clockwise otherwise Define a functor G w : k[x, x ]-mod Λ-mod as follows: If V is an object in k[x, x ]-mod, define G w (V ) to be the Λ-module which is defined by the representation of Q w, which assigns to each vertex of Q w the vector space V and to the arrow w the linear transformation x and to all other arrows the identity linear transformation Theorem Let Λ = kq/i be a string algebra Then the modules G w (V ), where w ranges over all strings and bands, and V = k if w is a string, and V ranges over all indecomposable k[x, x ]-modules if w is a band, form a complete set of representatives of indecomposable Λ-modules Proof See [7, Theorem on p 6] Definition Let Λ = kq/i be a string algebra (i) If C is a string, then we define the string module M(C) to be G C (k)

24 (ii) If B is a band, λ k, and n is a positive integer, then we define the band module M(B, λ, n) to be G B (V n (λ)) where V n (λ) is the k[x, x ]-module of k-dimension n on which x acts as the n n Jordan block J n (λ) Remark Given a special biserial algebra Λ = kq/i, we can construct a quotient algebra Λ of Λ which is a string algebra such that all non-projective indecomposable Λ-modules correspond to indecomposable Λ-modules by inflation More precisely, following [9, II], let L = {i Q 0 : Λe i is injective and not uniserial} and let S 0 = i L soc(λe i) Define Λ = Λ/S 0 Then Λ is a string algebra Moreover, the union of the set of isomorphism classes of all indecomposable Λ-modules (viewed as Λ-modules by inflation) with the set of isomorphism classes of Λe i, i L, coincides with the set of isomorphism classes of all indecomposable Λ-modules Therefore, when working with non-projective indecomposable Λ-modules, we can just consider indecomposable modules for Λ, which all arise as string and band modules Lemma 4 Suppose Λ is a special biserial algebra and Λ = Λ/S 0 as defined in Remark Then the following are equivalent: (i) Λ is of finite representation type (ii) Λ has finitely many strings (iii) Λ has no bands Proof See [9, Lemma II8]

25 4 5 Quivers and Relations for Brauer Tree Algebras Let Λ be a Brauer tree algebra with associated Brauer tree T (Λ) Following the methods outlined in [, Section X], we can construct from T (Λ) a quiver Q(Λ) and an ideal I of kq(λ) such that kq(λ)/i is isomorphic to the basic algebra of Λ We now demonstrate this with the examples from Section Example 5 Let Λ be the Brauer tree algebra from Example Then the quiver Q(Λ) is given as follows: α γ δ 4 µ β η Furthermore, the basic algebra of Λ is isomorphic to kq(λ)/i, where I =< αγβα, βαγ ηδ, δη µ, δβ, γη, µδ, ηµ > Example 6 Let Λ be the Brauer tree algebra associated to the star Brauer tree from Example 4 The quiver Q(Λ) is then as follows: e Labeling the arrow from i to i + by α i for i e and the arrow from e to by α e, the basic algebra of Λ is isomorphic to kq(λ)/i, where the ideal I is generated by all paths of length me +

26 5 We will also make use of the stable Auslander-Reiten quiver of a Brauer tree algebra For this reason, we now introduce the stable module category, Morita and stable equivalences, and Auslander-Reiten quivers Again, many of our definitions and results are taken from [5, Chapter ] 6 Projective Covers and the Stable Module Category Let Λ be an arbitrary finite dimensional k-algebra Recall that a projective cover of a Λ-module M is a projective Λ-module P together with an essential epimorphism f : P M, denoted by (P, f) Here, an epimorphism f : A B in Λ-mod is called an essential epimorphism if Ker(f) rad(a) By [, Theorem I4], every finitely generated Λ-module M has a projective cover (P, f) Moreover, any two projective covers (P, f ) and (P, f ) are isomorphic, in the sense that there exists a Λ-module isomorphism π : P P with f π = f Note that if M is a Λ-module, then (P, f) is a projective cover of M if and only if the epimorphism P/rad(P ) M/rad(M) induced by f is an isomorphism (see [, Proposition I4]) Definition 7 Let M be a finitely generated Λ-module The first syzygy of M, denoted by Ω(M), is the kernel of a projective cover f : P M By our discussion above, Ω(M) is unique up to isomorphism Definition 8 Let M and N be Λ-modules (i) Define P Hom Λ (M, N) to be the k-subspace of Hom Λ (M, N) consisting of those

27 6 Λ-module homomorphisms which factor through a projective Λ-module (ii) Define the k-vector space of stable homomorphisms from M to N to be the quotient space Hom Λ (M, N) = Hom Λ (M, N)/P Hom Λ (M, N) If M = N then Hom Λ (M, M) is a k-algebra denoted by End Λ (M) (iii) Define Λ-mod to be the category with Ob(Λ-mod)=Ob(Λ-mod) and morphisms Mor Λ-mod (M, N) = Hom Λ (M, N) for all objects M, N The category Λ-mod is called the stable module category Lemma 9 There exists a functor Ω : Λ-mod Λ-mod such that Ω(M) is the first syzygy for all Λ-modules M This functor is called the syzygy functor If Λ is self-injective, then Ω is an equivalence Proof See [, pp 4-6] Corollary 0 If Λ is a self-injective algebra, then End Λ (M) = End Λ (Ω(M)) for all finitely generated Λ-modules M Definition Let M and N be Λ-modules, and let δ P δ ɛ P P0 M 0 be a projective resolution of M Applying Hom Λ (, N) to the projective resolution of M, we obtain a sequence of k-vector spaces 0 Hom Λ (P 0, N) δ Hom Λ (P, N) δ Hom Λ (P, N) δ For all n 0, define Ext n Λ (M, N) = Ker(δ n+)/im(δ n) Since Hom Λ (, N) is left exact, Ext 0 Λ (M, N) = Ker(δ ) = Im(ɛ ) = Hom Λ (M, N)

28 7 Theorem If Λ is self-injective, and M and N are Λ-modules, then Ext n Λ(M, N) = Hom Λ (Ω n (M), N) for all n Proof See [0, Theorem 9] 7 Morita and Stable Equivalences Definition Let Λ and Λ 0 be finite dimensional k-algebras (i) We say Λ and Λ 0 are Morita equivalent, written Λ M Λ 0, if Λ-mod and Λ 0 -mod are equivalent categories (ii) We say Λ and Λ 0 are stably equivalent if Λ-mod and Λ 0 -mod are equivalent categories (iii) We say Λ and Λ 0 are stably equivalent of Morita type if there exists a Λ 0 -Λbimodule X and a Λ-Λ 0 -bimodule Y such that X and Y are projective as left and as right modules and Y Λ0 X = Λ P X Λ Y = Λ 0 Q as Λ-Λ-bimodules, and as Λ 0 -Λ 0 -bimodules, where P is a projective Λ-Λ-bimodule and Q is a projective Λ 0 -Λ 0 -bimodule In particular, X Λ and Y Λ0 induce mutually inverse equivalences between Λ-mod and Λ 0 -mod

29 8 Theorem 4 Two basic algebras Λ and Λ 0 are Morita equivalent if and only if they are isomorphic Proof See [9, Lemma I6] Theorem 5 There exists a unique basic algebra Λ 0 such that Λ M Λ 0 Proof See [9, Corollary I7] Note that Λ 0 is the same as in Definition 5 8 Almost Split Sequences and Auslander-Reiten Quivers Definition 6 Let A, B, C be Λ-modules (i) A morphism f : A B in Λ-mod is said to be left almost split if it is not a split monomorphism and any monomorphism A Y in Λ-mod which is not split factors through f (ii) A morphism g : B C in Λ-mod is said to be right almost split if it is not a split epimorphism and any epimorphism X C in Λ-mod which is not split factors through g (iii) An exact sequence 0 A f B g C 0 in Λ-mod is called an almost split sequence if f is left almost split and g is right almost split Theorem 7 (a) If C is an indecomposable non-projective Λ-module, then there exists an almost split sequence 0 A f B g C 0 in Λ-mod (b) If A is an indecomposable non-injective Λ-module, then there exists an almost split sequence 0 A f B g C 0 in Λ-mod

30 9 Proof See [, Theorem V5] Definition 8 Suppose 0 A f B g C 0 is an almost split sequence in Λ-mod Define the Auslander translate τc of C to be τc = A, and define τ A = C Proposition 9 If Λ is symmetric, then for each non-projective Λ-module C we have τc = Ω (C) Proof See [, Proposition IV8] Definition 40 A morphism g : B C in Λ-mod is called irreducible if it is neither a split monomorphism nor a split epimorphism, and if g = t s for certain s : B X and t : X C in Λ-mod, then s is a split monomorphism or t is a split epimorphism Theorem 4 Let A and C be indecomposable Λ-modules (i) A morphism f : A B in Λ-mod is irreducible if and only if there exists a ( ff morphism f : A B in Λ-mod such that the induced morphism ) : A B B is a minimal left almost split morphism (ii) A morphism g : B C in Λ-mod is irreducible if and only if there exists a morphism g : B C in Λ-mod such that the induced morphism (g, g ) : B B C is a minimal right almost split morphism Proof See [, Theorem V5] Definition 4 (i) The Auslander-Reiten quiver of Λ is the quiver Γ(Λ) whose vertices are the isomorphism classes of indecomposable Λ-modules Denoting

31 0 the vertex corresponding to an indecomposable Λ-module M by [M], there is an arrow [M] [N] between two vertices if and only if there is an irreducible morphism M N (ii) The stable Auslander Reiten quiver of Λ is the quiver Γ S (Λ) which is obtained from Γ(Λ) by removing all vertices [τ i P ] and [τ i I] and all adjacent arrows for all projective Λ-modules P and all injective Λ-modules I, and all i 0 In the special case when Λ is self-injective, one only has to remove the vertices [P ] for P projective and the adjacent arrows Remark 4 Suppose Λ = kq/i is a basic algebra Define L = {i Q 0 Λe i is injective} and let S = i L soc(λe i) Then the isomorphism classes of all indecomposable Λ-modules are given by the isomorphism classes of all indecomposable Λ/S-modules together with the isomorphism classes of Λe i, i L Moreover, the Auslander-Reiten quiver Γ(Λ/S) is obtained from Γ(Λ) by removing the vertices [Λe i ], i L If Λ is self-injective, then S = soc(λ) and Γ(Λ/S) is the stable Auslander-Reiten quiver Γ S (Λ) (see [9, I8]) Lemma 44 Let F : Λ-mod Λ -mod be a stable equivalence between self-injective algebras If Λ and Λ have no block of radical length, then Γ S (Λ) and Γ S (Λ ) are isomorphic stable translation quivers Proof See [, Corollary X9]

32 9 Homomorphisms and Almost Split Sequences for String Modules Let Λ = kq/i be a special biserial algebra By Remark, we have a string algebra Λ, which is a quotient algebra of Λ, such that all non-projective indecomposable Λ-modules correspond to indecomposable Λ-modules by inflation Furthermore, we have that the Auslander-Reiten quiver Γ(Λ) of Λ is obtained from Γ(Λ) by removing all vertices corresponding to non-uniserial indecomposable Λ-modules that are both projective and injective (see [9, II]) Therefore, to construct the homomorphisms between the indecomposable non-projective Λ-modules, it is enough to find the homomorphims between the indecomposable Λ-modules Definition 45 Let S and T be strings for Λ Suppose C is a substring of both S and T such that the following conditions are satisfied: (i) S BCD, where B is a substring which is either of length 0 or B = B τ for an arrow τ, and D is a substring which is either of length 0 or D = σ D for an arrow σ In other words, S B τ C σ D (ii) T ECF, where E is a substring which is either of length 0 or E = E ν for an arrow ν, and F is a substring which is either of length 0 or F = µf for an arrow µ In other words, T E ν C µ F

33 Then there exists a canonical Λ-module homomorphism α C : M(S) M(C) M(T ) Theorem 46 Each f Hom Λ (M(S), M(T )) can be written uniquely as a k- linear combination of canonical Λ-module homomorphisms as in Definition 45 In particular, if M(S) = M(T ), then the canonical endomorphisms form a k-basis of End Λ (M(S)) Proof See [] Definition 47 Let C = c c c n be a string for Λ of length n 0 (i) We say that C starts on a peak if there is no arrow β such that Cβ is a string, and we say that C ends on a peak if there is no arrow β such that β C is a string (ii) We say that C starts in a deep if there is no arrow γ such that Cγ is a string, and we say that C ends in a deep if there is no arrow γ such that γc is a string (iii) We call C a directed string if all c j are arrows, and we call C an inverse string if all c j are formal inverses of arrows Definition 48 Let C be a string for Λ (i) If C does not start on a peak, then there is an arrow β such that Cβ is a string There is a unique directed string D such that CβD is a string starting in a deep We use the notation C h = CβD and say C h is obtained from C by adding a hook on the right

34 (ii) If C does not end on a peak, then there is an arrow β such that β C is a string There is a unique directed string D such that Dβ C is a string ending in a deep We use the notation h C = Dβ C and say h C is obtained from C by adding a hook on the left (iii) If C does not start in a deep, then there is an arrow γ such that Cγ is a string There is a unique directed string D such that Cγ D is a string starting on a peak We use the notation C c = Cγ D and say C c is obtained from C by adding a cohook on the right (iv) If C does not end in a deep, then there is an arrow γ such that γc is a string Then there is a unique directed string D such that D γc is a string ending on a peak We use the notation c C = D γc and say c C is obtained from C by adding a cohook on the left Definition 49 We call the following exact sequences of Λ-modules canonical exact sequences: (i) Let α be an arrow starting at vertex i The almost split sequence ending in V α = Λe i /Λα is of the form 0 τ(v α ) N α V α 0 where N α is a string module M(S) such that S is a string of the form S = D βc and C and D are maximal directed strings (See [7, Section ]) (ii) If C is a string which neither starts nor ends on a peak, then C h, h C, and h C h

35 4 exist, and we have an exact sequence 0 M(C) M( h C) M(C h ) M( h C h ) 0 (iii) If C is a string which does not start on a peak but ends on a peak, then C h exists We can write C = c D for some string D not starting on a peak, and hence D h exists We have an exact sequence 0 M(C) M(D) M(C h ) M(D h ) 0 (iv) If C is a string which starts on a peak but does not end on a peak, then h C exists We can write C = D c for some string D not ending on a peak, and hence hd exists We have an exact sequence 0 M(C) M( h C) M(D) M( h D) 0 (v) If C is a string which both starts on a peak and ends on a peak, then we can write C = c D c for some string D We have an exact sequence 0 M(C) M(D c ) M( c D) M( c D c ) 0 Theorem 50 The canonical exact sequences are the almost split sequences of Λ- modules containing string modules Proof See [7, Proposition on p 7] 0 The Stable Auslander-Reiten Quiver for Brauer Tree Algebras For this section, let Λ be a Brauer tree algebra with related Brauer tree T (Λ), which has e edges and multiplicity m

36 5 Recall from Proposition 7 that Λ is a symmetric special biserial algebra of finite representation type, which means by Lemma 4 that Λ has finitely many strings and no bands Furthermore, the description of the almost split sequences between string modules in Section 9 shows that the stable Auslander-Reiten quiver Γ S (Λ) is a finite tube ZA me / τ e = ZA me / Ω e : e- e me- me where we identify the left and right edge Note that by Definition 49 and Theorem 50, the vertices [H] of Γ S (Λ) that have precisely one predecessor and one successor correspond to either uniserial nonprojective modules H of maximal length or to simple modules H such that the edge in T (Λ) corresponding to H has a non-exceptional leaf vertex Definition 5 Let Λ be a Brauer tree algebra with stable Auslander-Reiten quiver Γ S (Λ), and let M and N be indecomposable Λ-modules Note that τ = Ω in this case (i) Let l and let (M = X 0, X,, X l = N) be a sequence of successively nonisomorphic Λ-modules in Γ S (Λ) such that for i l, there is an irreducible

37 6 morphism f i : X i X i Then we call f := f l f a hom-path of length l from M to N When l = 0, a hom-path f of length 0 from M to M corresponds to the sequence (M) and is given by an automorphism of M We call a hom-path directed if it does not contain any subpath from τ(x) to X for any module X (ii) A row of Γ S (Λ) containing [M] is defined to be the set R containing [τ i (M)] for all integers i (iii) We say M is a module on a boundary of Γ S (Λ) if [M] has only one predecessor and only one successor in Γ S (Λ) In this case, we call the row R containing [M] a boundary of Γ S (Λ) Note that if e = and m > or if e > then there are precisely two boundaries of Γ S (Λ) (iv) We say M is distance l from a given row R of Γ S (Λ) if l is the minimal length among all hom-paths from M to any module in R Proposition 5 Let Γ S (Λ) be the stable Auslander-Reiten quiver for Λ If e = and m >, there is exactly one vertex at each of the two boundaries of Γ S (Λ) Suppose now that e > Let H be a module on a boundary of Γ S (Λ) with E =soc(h) and E =top(h) Let W H = (v, X, v,, X e, v e+ ) be a complete clockwise walk of multiplicity around T (Λ) with X = E and X = E A uniserial module H of maximal length with E =soc(h ) and E =top(h ) belongs to the same boundary of Γ S (Λ) as H if and only if there exists j e with E = X j and E = X j Proof See [4, Proposition 7]

38 7 Theorem 5 Let M and N be two non-projective indecomposable Λ-modules Let d (resp d ) be the length of a shortest hom-path from Ω (N) to M (resp from M to N) Then dim k Hom Λ (M, N) = max(0, s + ), where s Z satisfies the equation: me = d + d + es Proof See [4, Theorem 5] Corollary 54 Supposed M and N are non-projective indecomposable Λ-modules and i > 0 Let d (resp d ) be the length of a shortest hom-path from Ω (N) to Ω i (M) (resp from Ω i (M) to N) Then dim k Ext i Λ (M, N) = max(0, s + ), where s Z satisfies the equation: me = d + d + es Proof See [4, Corollary 54] Example 55 Consider the Brauer tree algebra Λ from Example and label the related Brauer tree as in Example 6 Using the quiver with relations of the basic algebra constructed in Example 5, and the methods from Sections 4 and 9, we construct the stable Auslander-Reiten quiver Γ S (Λ) in Figure To illustrate Proposition 5, consider the Λ-module H =, which is the uniserial module of length with top(h) = and soc(h) = This module is on the upper boundary of Γ S (Λ) The related walk from Proposition 5 is W H = (v,, v,, v 4, 4, v 5, 4, v 4,, v,, v,, v,, v ), and it produces all uniserial modules that are seen on the upper boundary of Γ S (Λ) in Figure Example 56 Consider the Brauer tree algebra Λ from Example 4 with e = m =

39 Figure : Stable Auslander-Reiten quiver for Λ in Example

40 9 Using the quiver and relations of the basic algebra found in Example 6, we construct the stable Auslander-Reiten quiver Γ S (Λ) in Figure To illustrate Theorem 5, consider the module M =, which is the uniserial module of length 4 with top(m) = Ω (M) = To find dim k Hom Λ (M, M), first note that, which is uniserial of length 6 with top(m) = Then using Γ S (Λ) in Figure, we can see that d = and d = 0, so we get that 8 = me = d + d + es = + 6s Thus s = and dim k Hom Λ (M, M) = max(0, s + ) = The main goal of this thesis is to determine versal deformation rings of certain modules for Brauer tree algebras For this reason, we now give an introduction to versal deformation rings Many of our definitions and results are taken from [5] and [6] Versal Deformation Rings Let Ĉ be the category of all complete local commutative Noetherian k-algebras with residue field k, where the morphisms are continuous k-algebra homomorphisms inducing the identity on k Let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module Then V is finite dimensional as a k-vector space, say dim k V = n which means that we can identify V = k n as a k-vector space Moreover, the Λ-module structure of V is given by the k-algebra homomorphism Λ α V End k (V ) = Mat n n (k) with α V (x) = A x for all x Λ, where A x v = xv for all v V Let R Ob(Ĉ) and let π : R k be the natural surjection onto the residue field Define RΛ = R k Λ

41 0 Figure : Stable Auslander-Reiten quiver for Λ in Example 4

42 Definition 57 (i) A lift of V over R is a free R-module M = R n, where n = dim k V as above, together with a k-algebra homomorphism Λ α M End R (M) = Mat n n (R) such that π α M = α V Put differently, a lift of V over R is a finitely generated RΛ-module M, which is free as an R-module, together with a Λ-module isomorphism φ : k R M V We denote this lift of V by (M, φ) (ii) Two lifts (M, φ) and (M, φ ) of V over R are isomorphic if there exists an RΛ-module isomorphism f : M M such that φ = φ (id f) (iii) The isomorphism class of the lift (M, φ) is called a deformation of V over R and it is denoted by [M, φ] The set of deformations of V over R is denoted by Def Λ (V, R) Definition 58 We define a covariant functor F : Ĉ Sets as follows: For R Ob(Ĉ), let F (R) = Def Λ(V, R) For α : R R in Ĉ, define F (α) : Def Λ(V, R) Def Λ (V, R ) by F (α)([m, φ]) = [R R,α M, φ α ] where φ α is the composition k R (R R,α M) We call F the deformation functor associated to V = k R M φ V Theorem 59 Let F be the deformation functor associated to V Then there exists a ring R(Λ, V ) in Ĉ and a deformation [U, φ U] of V over R(Λ, V ) such that for every R in Ĉ and every lift (M, φ) of V over R, there exists a morphism α : R(Λ, V ) R in Ĉ, which is not necessarily unique, such that F (α)([u, φ U]) = [M, φ] Moreover,

43 in the case that R is the ring of dual numbers k[ɛ] = k[u]/(u ), the morphism α is unique Proof See [5, Proposition and Remark ] Definition 60 In the situation of Theorem 59, we call R(Λ, V ) the versal deformation ring of V and [U, φ U ] the versal deformation of V over R(Λ, V ) If F is representable, then there exists a ring R(Λ, V ) in Ĉ and a deformation [U, φ U ] of V over R(Λ, V ) such that for every R in Ĉ and every lift (M, φ) of V over R, there exists a unique morphism α : R(Λ, V ) R in Ĉ such that F (α)([u, φ U]) = [M, φ] In this case, we call R(Λ, V ) the universal deformation ring of V and [U, φ U ] the universal deformation of V over R(Λ, V ) Theorem 6 Let Λ be a finite dimensional k-algebra such that Λ and Λ are Morita equivalent If the Λ-module V corresponds to the Λ -module V under this Morita equivalence, then R(Λ, V ) = R(Λ, V ) Proof See [5, Proposition 5] Theorem 6 Let F be the deformation functor associated to V Then F is representable if one of the following is true: (i) End Λ (V ) = k, or (ii) Λ is self-injective and End Λ (V ) = k Proof See [5, Proposition and Theorem 6(ii)]

44 Theorem 6 Let Λ be a Frobenius algebra, and let V be a Λ-module which has no projective direct summands Then R(Λ, V ) = R(Λ, Ω(V )) Proof The case where the stable endomorphism ring of V is isomorphic to k has been proved in [6, Section 6] and [5, Theorem 6] Following the argumentation in [6, Section 6], we only have to ensure that we can replace the arguments involving [6, Lemma 0] We fix a projective cover ɛ : P (V ) V of V, such that Ω(V ) is the kernel of ɛ Since Λ is self-injective, it follows that P (V ) is an injective Λ-module Let R be an Artinian object in Ĉ We first prove two claims, using the following set-up Set-up For i =,, let X i be a finitely generated RΛ-module which is free as an R-module such that there is a Λ-module isomorphism ξ i : k R X i V Assume that there exists a short exact sequence of RΛ-modules ι 0 Y i ψ i i Pi Xi 0 such that P i is a projective RΛ-module with k R P i = P (V ) Let Ω(ξi ) be any Λ- module homomorphism such that there exists a commutative diagram of Λ-modules of the form: id ι 0 i id ψ k R Y i i k R P i k R X i 0 Ω(ξ i ) 0 Ω(V ) Ξ i ι P (V ) ξ i ɛ V 0

45 4 Note that since ɛ is an essential epimorphism, Ξ i must be surjective, and hence a Λ-module isomorphism In particular, every such Ω(ξ i ) is a Λ-module isomorphism Claim For i =,, suppose X i, Y i, P i, ξ i, ι i, ψ i, Ω(ξ i ) are as in the Set-up If there exists an RΛ-module isomorphism ν : X X with ξ (id ν) = ξ then there exists an RΛ-module isomorphism µ : Y Y with Ω(ξ ) (id µ) = Ω(ξ ) Proof of Claim Consider the pullback Q of ψ and ν ψ Since ν is an isomorphism, we obtain the following commutative diagram of RΛ-modules with exact rows and columns 0 Ker(l ) κ 0 Y () 0 Ker(l ) j j Q ι l P 0 κ l 0 Y ι P ν ψ X ψ where, for i =,, j i is the inclusion homomorphism and κ i is an isomorphism For i =,, let m i : P i Q be an RΛ-module homomorphism such that l i m i is the identity on P i Since k R X i = V has no projective direct summands by assumption

46 5 and since Λ is self-injective, k R Y i = Ω(V ) also has no projective direct summands Hence Y i has no projective direct summands Decomposing Y and Y into a direct sum of indecomposable RΛ-modules and using the same arguments as in the proof of the Krull-Schmidt-Azumaya Theorem (see [8, Theorem 6]), it follows that l m is an RΛ-module isomorphism Letting λ = l m, we obtain that ψ λ = ν ψ Hence there exists an RΛ-module homomorphism µ : Y Y giving a commutative diagram of RΛ-modules ι 0 Y ψ P X 0 () µ λ 0 Y ι P ψ X 0 ν In particular, µ has to be an RΛ-module isomorphism Since Λ is self-injective, Ω defines an autoequivalence of Λ-mod Because ξ (id ν) = ξ in Λ-mod, this implies that Ω(ξ ) (id µ) = Ω(ξ ) in Λ-mod This means that there exists a Λ-module homomorphism p : k R Y k R Y such that p factors through a projective Λ-module and Ω(ξ ) (id µ + p) = Ω(ξ ) in Λ-mod in Λ-mod Since finitely generated projective Λ-modules are injective, it follows that p factors through id ι, say p = q (id ι ) where q : k R P k R Y Because P is a projective RΛ-module, there exists an RΛ-module homomorphism q R : P Y

47 6 such that id q R = q Using Diagram (), we obtain a commutative diagram of RΛ-modules ι 0 Y ψ P X 0 such that µ+q R ι λ+ι q R 0 Y ι P ψ X 0 ν Ω(ξ ) (id ( µ + q R ι )) = Ω(ξ ) in Λ-mod Recall that λ = l m where m : P Q is an RΛ-module homomorphism such that l m is the identity on P Let now m = m + j κ q R Using Diagram (), we see that l m = l ( m + j κ q R ) = l m is the identity on P, and λ + ι q R = l m + l j κ q R = l m Hence, letting λ = λ + ι q R = l m we see, as in the proof of the Krull-Schmidt-Azumaya Theorem, that λ is an RΛmodule isomorphism Therefore, µ = µ + q R ι is also an RΛ-module isomorphism Since Ω(ξ ) (id µ) = Ω(ξ ), this proves Claim For any RΛ-module M, define M = Hom R (M, R) where the right RΛ-module structure of M is induced by the left RΛ-module structure of M Since Λ is a

48 7 Frobenius algebra, it follows for i =, that P i is a projective right RΛ-module (see Claim 7 of the proof of [5, Theorem 6]) If M, N are finitely generated RΛ-modules which are free over R and σ : M N is an RΛ-module homomorphism, then we can (and will) identify k R M = (k R M) and id σ = (id σ) Claim For i =,, suppose X i, Y i, P i, ξ i, ι i, ψ i, Ω(ξ i ) are as in the Set-up If there exists an RΛ-module isomorphism µ : Y Y with Ω(ξ ) (id µ) = Ω(ξ ) then there exists an RΛ-module isomorphism ν : X X with ξ (id ν) = ξ Proof of Claim The assumptions imply that µ : Y Y is an isomorphism of right RΛ-modules satisfying (id µ ) Ω(ξ ) = Ω(ξ ) As in the proof of Claim, we see that Ω(V ) has no projective direct summands Because Λ is a Frobenius algebra, it follows that the k-dual of a projective Λ-module is a projective right Λ-module, which implies that Ω(V ) also has no projective direct summands Similarly, since (Xi ) = Xi as RΛ-modules, for i =,, and k R X i = V, X i has no projective direct summands (see Claim 7 of the proof of [5, Theorem 6]) Therefore, we can use the same argumentation as in the proof of Claim to

49 8 see that there exists an isomorphism of right RΛ-modules τ : X X such that (id τ) ξ = ξ This implies ξ (id τ ) = ξ Note that for each RΛ-module M that is free over R, the map d M : M M, defined by d M (x)(f) = f(x) for all x M and f M, is an RΛ-module isomorphism Moreover, if σ : M N is an RΛ-module homomorphism, then σ = d N σ (d M ) Hence it follows that τ : X X defines an RΛ-module isomorphism ν : X X by ν = (d X ) τ d X such that ξ (id ν) = ξ, which proves Claim To prove Theorem 6, we now follow the arguments in [6, Section 6], where in all statements we omit the assumption that the stable endomorphism ring End Λ (V ) is isomorphic to k More precisely, the proof of [6, Lemma 6] goes through without any change To show the map g Ω,R : F V (R) F Ω(V ) (R) in [6, Equation (7)] is well-defined, we use Claim instead of [6, Lemma 0] In the proof of [6, Lemma 6], we again use Claim instead of [6, Lemma 0] The proofs of [6, Lemmas 64 and 65] go through without any change We use Claim to prove [6, Lemma 66] Therefore, we obtain Theorem 6 in the same way as [6, Theorem 67] Theorem 64 Suppose F is the deformation functor associated to V If dim k Ext Λ(V, V ) = r,

50 9 then there exists a surjective homomorphism λ : k[[t,, t r ]] R(Λ, V ) in Ĉ, and r is minimal with this property Proof Let k [ɛ] be the ring of dual numbers, ie k [ɛ] = k [u] /(u ) Because there is a unique morphism α : R(Λ, V ) k [ɛ] in Ĉ for each deformation of V over k [ɛ], it follows that F (k [ɛ]) = HomĈ(R(Λ, V ), k [ɛ]) Since by [5, Proposition ], F (k [ɛ]) = Ext Λ (V, V ) as a k-vector space, this implies the theorem

51 40 CHAPTER VERSAL DEFORMATION RINGS OF MODULES FOR BRAUER TREE ALGEBRAS For the entirety of this chapter, k is an algebraically closed field of arbitrary characteristic, and Λ is a Brauer tree algebra with associated Brauer tree T (Λ), which has e edges and multiplicity m We assume e > or m > since if e = = m there is only one non-projective indecomposable Λ-module S up to isomorphism, and S is simple The goal of this thesis is to determine the versal deformation rings for the indecomposable Λ-modules M In the case where End Λ (M) = k, M has been shown to have a universal deformation ring, and these rings have been determined in [] In this thesis, we will calculate the versal deformation ring for M in the case where dim k End Λ (M) > Dimensions of Ext Λ (M, M) and End Λ(M) for an indecomposable Λ-module M Let Γ S (Λ) be the stable Auslander-Reiten quiver of Λ and let M be an indecomposable non-projective Λ-module It was shown in [4] that the dimensions of End Λ (M) and Ext Λ (M, M) depend only on the relative locations of [M], [Ω (M)] and [Ω(M)] in Γ S (Λ) (see Theorem 5 and Corollary 54) In this section, we will show how to determine these dimensions from the distance of M to the closest boundary of Γ S (Λ)