Universal deformation rings of modules over self-injective algebras

Let $k$ be a field, and let $\Lambda$ be a finite dimensional $k$-algebra. We prove that if $\Lambda$ is a self-injective algebra, then every finitely generated $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(\Lambda,V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $\Lambda$ is also a Frobenius algebra, we show that $R(\Lambda,V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra $\Lambda_0$ of dihedral type, as introduced by Erdmann, and we determine $R(\Lambda_0,V)$ for every finitely generated $\Lambda_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.Comment: 25 pages, 2 figures. Some typos have been fixed, the outline of the paper has been changed to improve readabilit

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“…Part of this paper constitutes the Ph.D. thesis of the second author under the supervision of the first author [19].…”

Section: Introductionmentioning

confidence: 99%

Universal deformation rings of modules over Frobenius algebras

Bleher

Vélez-Marulanda

2012

Journal of Algebra

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“…For this reason, we now give an introduction to versal deformation rings. Many of our definitions and results are taken from [5] and [16].…”

Section: Morita and Stable Equivalencesmentioning

confidence: 99%

Versal deformation rings of modules over Brauer tree algebras

Wackwitz¹

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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 2). 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 1 Λ

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“…Many of our definitions, results and proof ideas are based on work by Schlessinger [10] and Mazur [8] and on Vélez-Marulanda's thesis [12]. The main difference is that [10], [8] and [12] worked withĈ comm instead ofĈ. This requires us to modify their definitions, results and proof ideas to fit our situation.…”

Section: Chapter 3 Non-commutative Deformation Ringsmentioning

confidence: 99%

“…Noting that e 22 has yet to be defined, we choose to set e 22 = 0. Then we define e 11 as e 11 = e 22 − l 12 = −l 12 , which we obtain by noticing that e 22 − e 11 = (XE − EX) 12 .…”

Section: 1mentioning

confidence: 99%

Non-commutative deformation rings

Margolin¹

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ACKNOWLEDGEMENTSExistence and uniqueness are more than abstract mathematical concepts.They are human concepts as well. While this thesis proves powerful mathematical results, it is perhaps more significant to note that this thesis is proof of the power of family, persistence, and the notion that unsolved problems are not necessarily ones that are impossible to solve.While this thesis was physically written by me, I am a mere asterisk in the list of its contributors. For existence, uniqueness, and everything in between, I thank my parents, Mom and Dad, whose support is one of the confounding and wonderful manifestations of the infinite. Not of any lesser cardinality, I want to thank my sisters, Lisa and Becky, for always making sure I knew that I was never alone in solving the problems of life.To my wife, Adri, you are the reason I went back to school. Far more importantly, however, you are proof that dreams are worth dreaming, and for that I am forever thankful. I love you. I owe Professor Dan Anderson a very special kind of gratitude, as he provided me with a solution to the problem of second chances. During a time when my lifestyle might have been converging to unenviable values, you opened up a neighborhood of knowledge, support and trust which allowed me to redefine myself in ways which I had once forgotten existed. None of this would have happened without your willingness to take a chance on me. Thank you.ii To Professor Frauke Bleher, what can I begin to say. That is a sentence which expresses a statement of fact, so please do not try and amend the punctuation, as you have made sufficiently many corrections to my thesis, already, all of which I am thankful for. Without your knowledge and expertise, I am not ashamed to say that my fingerprint on this subject would not have found the light of day. Your willingness to take me on as a student in trying times-for multiple reasons-was both motivating and humbling. You are an amazing advisor, an amazing supporter and defender of my work and my life interests, but above all you are an amazing teacher. I express profound gratitude to you for the time I have been able to learn from you, and I thank you for always having my best interests at heart.At the end of the day, this thesis is dedicated to the subject of Mathematics. Whenever a mathematician is asked what usefulness his or her project or work provides, may he or she confidently respond that it likely addresses either a current intellectual need or is in advance of a future one, and perhaps even provides its inventor with a non-zero amount of happiness. What else could we hope for?iii ABSTRACT The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.The thesis is divided into two part...

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Universal deformation rings of modules over self-injective algebras

Cited by 3 publications

References 37 publications

Universal deformation rings of modules over Frobenius algebras

Universal deformation rings of modules over Frobenius algebras

Versal deformation rings of modules over Brauer tree algebras

Non-commutative deformation rings

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