Non-commutative deformation rings

Abstract

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...