#### Document Type

Dissertation

#### Date of Degree

Spring 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Frauke Bleher

#### 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 parts.

In the first part, *A* is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field *k*. A category Ĉ is defined whose objects are complete local λ-algebras *R* with residue field *k* such that *R* is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If *V* is a finite dimensional *k*-vector space that is also a left *A*-module and that satisfies a natural finiteness condition, it is proved that *V* has a so-called versal deformation ring *R(A,V)*. More precisely, *R(A,V)* is an object in Ĉ such that the isomorphism class of every lift of *V* over an object *R* in Ĉ arises from a morphism α : *R(A,V)*→ *R* in Ĉ and α is unique if *R* is the ring of dual numbers *k*[ϵ].

In the second part, two particular examples of λ, *A* and *V* are studied and the versal deformation ring *R(A,V)* is determined in each of these cases. In the first example, λ=*k*, *A* is a series of non-commutative *k*-algebras depending on a parameter *r≥2*, and *V* is a particular quotient module of *A*; it is shown that *R(A,V)* is isomorphic to *A*. The second example builds on the first example when *r=2* and uses that, if additionally the characteristic of *k* is 2, then *A* is isomorphic to the group ring *k[D _{8}]* of a dihedral group

*D*of order 8.

_{8}It is shown that if *k* is perfect and *W* is the ring of infinite Witt vectors over *k*, then *R(W[D _{8}],V)* is isomorphic to

*W[D*.

_{8}]#### Public 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 parts.

In the first part, a theory of deformations of representations of an algebra *A* over a certain class of complete local rings with a fixed residue field *k* is developed. It is shown that every finite dimensional *k*-vector space *V* that is also a left *A*-module and that satisfies a natural finiteness condition has a so-called versal deformation ring *R*(*A, V*), which is such a complete local ring with residue field *k*.

In the second part, two particular examples of non-commutative algebras *A* and modules *V* are studied and the versal deformation ring *R*(*A, V*) is shown to be isomorphic to *A* in each of these cases.

#### Keywords

publicabstract

#### Pages

viii, 128 pages

#### Bibliography

Includes bibliographical references (page 128).

#### Copyright

Copyright 2016 Benjamin Paul Margolin