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Northeastern University

1. Russell, Jeremy. A functorial approach to linkage and the asymptotic stabilization of the tensor product.

Degree: PhD, Department of Mathematics, 2013, Northeastern University

URL: http://hdl.handle.net/2047/d20003138

This thesis consists of three projects.; The first project deals with generalizing the definition of zeroth derived functors to work for any abelian category. The classical definitions of zeroth derived functors require existence of injectives or pro jectives. In this project, we give definitions of the zeroth derived functors that do not require the existence of injectives or projectives. The new definitions result in generalized definitions of projective and injective stabilization of functors. The category of coherent functors is shown to admit a zeroth right derived functor. An interesting result of this fact is a counterpart to the Yoneda lemma for coherent functors. Moreover, zeroth derived functors are seen more appropriately as approximations of functors by left exact or right exact functors. Under certain reasonable conditions, the category of coherent functors is shown to have enough injectives. This result was first shown by Ron Gentle. We give an alternate proof of this fact.; The second project deals with extending the definition of horizontally linked modules over semiperfect Noetherian rings to the category of finitely presented functors over arbitrary Noetherian rings. Linkage of modules can be defined using the syzygy and transpose operation. Auslander established the merit of studying modules by studying the functors from the module category into the category of abelian groups. It turns out that the module theoretic notion of linkage can be extended to a functorial notion of linkage and the satellite endofunctors are crucial to this extension.; The third project deals with finding alternate ways of recovering Vogel homology. There are three known ways of generalizing Tate cohomology to a cohomology theory that works over arbitrary rings, that given by Vogel, that given by Mislin, and that given by Buchweitz. Vogel also provided a homological counterpart to his generalization of Tate cohomology. Yoshino attempted to recover this homology theory using an approach similar to Mislin's approach to recovering Tate cohomology; however, he only produced Vogel homology in positive degrees. This is fixed by returning to Mislin's construction and observing that it can be dualized. Completely missing from the picture was an approach similar to Buchweitz's approach to generalizing Tate cohomology. The asymptotic stabilization of the tensor product is introduced to fill this gap.

Subjects/Keywords: coherent functors; linkage; satellites; Vogel (co)homology; zeroth derived functors; Mathematics

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APA (6^{th} Edition):

Russell, J. (2013). A functorial approach to linkage and the asymptotic stabilization of the tensor product. (Doctoral Dissertation). Northeastern University. Retrieved from http://hdl.handle.net/2047/d20003138

Chicago Manual of Style (16^{th} Edition):

Russell, Jeremy. “A functorial approach to linkage and the asymptotic stabilization of the tensor product.” 2013. Doctoral Dissertation, Northeastern University. Accessed October 28, 2020. http://hdl.handle.net/2047/d20003138.

MLA Handbook (7^{th} Edition):

Russell, Jeremy. “A functorial approach to linkage and the asymptotic stabilization of the tensor product.” 2013. Web. 28 Oct 2020.

Vancouver:

Russell J. A functorial approach to linkage and the asymptotic stabilization of the tensor product. [Internet] [Doctoral dissertation]. Northeastern University; 2013. [cited 2020 Oct 28]. Available from: http://hdl.handle.net/2047/d20003138.

Council of Science Editors:

Russell J. A functorial approach to linkage and the asymptotic stabilization of the tensor product. [Doctoral Dissertation]. Northeastern University; 2013. Available from: http://hdl.handle.net/2047/d20003138

University of Michigan

2. Wiltshire-Gordon, John D. Representation Theory of Combinatorial Categories.

Degree: PhD, Mathematics, 2016, University of Michigan

URL: http://hdl.handle.net/2027.42/133178

A representation V of a category D is a functor D – > Mod-R; the representations of D form an abelian category with natural transformations as morphisms. Say V is finitely generated if there exist finitely many vectors v_i in V d_i so that any strict subrepresentation of V misses some v_i. If every finitely generated representation satisfies both ACC and DCC on subrepresentations, we say D has dimension zero over R. The main theoretical result of this thesis is a practical recognition theorem for categories of dimension zero (Theorem 4.3.2). The main computational result is an algorithm for decomposing a finitely presented representation of a category of dimension zero into its multiset of irreducible composition factors (Theorem 4.3.5). Our main applications take D to be the category of finite sets; we explain how the general results of this thesis suggest specific experiments that lead to structure theory and practical algorithms in this case.
*Advisors/Committee Members: Speyer, David E (committee member), Shi, Yaoyun (committee member), Snowden, Andrew (committee member), Fulton, William (committee member), Lagarias, Jeffrey C (committee member).*

Subjects/Keywords: Representation theory of categories; Representation stability; Categories of dimension zero; Coherent functors; Mathematics; Science

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wiltshire-Gordon, J. D. (2016). Representation Theory of Combinatorial Categories. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/133178

Chicago Manual of Style (16^{th} Edition):

Wiltshire-Gordon, John D. “Representation Theory of Combinatorial Categories.” 2016. Doctoral Dissertation, University of Michigan. Accessed October 28, 2020. http://hdl.handle.net/2027.42/133178.

MLA Handbook (7^{th} Edition):

Wiltshire-Gordon, John D. “Representation Theory of Combinatorial Categories.” 2016. Web. 28 Oct 2020.

Vancouver:

Wiltshire-Gordon JD. Representation Theory of Combinatorial Categories. [Internet] [Doctoral dissertation]. University of Michigan; 2016. [cited 2020 Oct 28]. Available from: http://hdl.handle.net/2027.42/133178.

Council of Science Editors:

Wiltshire-Gordon JD. Representation Theory of Combinatorial Categories. [Doctoral Dissertation]. University of Michigan; 2016. Available from: http://hdl.handle.net/2027.42/133178

University of Kansas

3. Se, Tony. Depth and Associated Primes of Modules over a Ring.

Degree: PhD, Mathematics, 2016, University of Kansas

URL: http://hdl.handle.net/1808/21895

This thesis consists of three main topics. In the first topic, we let R be a commutative Noetherian ring, I,J ideals of R, M a finitely generated R-module and F an R-linear covariant functor. We ask whether the sets \operatorname{Ass}_{R} F(M/I^{n} M) and the values \operatorname{depth}_{J} F(M/I^{n} M) become independent of n for large n. In the second topic, we consider rings of the form R = k[x^{a},x^{p1}y^{q1}, …,x^{pt}y^{qt},y^{b}], where k is a field and x,y are indeterminates over k. We will try to formulate simple criteria to determine whether or not R is Cohen-Macaulay. Finally, in the third topic we introduce and study basic properties of two types of modules over a commutative Noetherian ring R of positive prime characteristic. The first is the category of modules of finite F-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call F-abundant modules. These include, for example, the ring R itself and the canonical module when R has positive splitting dimension. We prove many facts about these two categories and how they are related. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest, such as complete intersections and invariant subrings.
*Advisors/Committee Members: Dao, Hailong (advisor), Jiang, Yunfeng (cmtemember), Katz, Daniel (cmtemember), Lang, Jeffrey (cmtemember), Nutting, Eileen (cmtemember).*

Subjects/Keywords: Mathematics; Cohen-Macaulay; coherent functors; divisor class group; F -regularity; Frobenius endomorphism; semigroup rings

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Se, T. (2016). Depth and Associated Primes of Modules over a Ring. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/21895

Chicago Manual of Style (16^{th} Edition):

Se, Tony. “Depth and Associated Primes of Modules over a Ring.” 2016. Doctoral Dissertation, University of Kansas. Accessed October 28, 2020. http://hdl.handle.net/1808/21895.

MLA Handbook (7^{th} Edition):

Se, Tony. “Depth and Associated Primes of Modules over a Ring.” 2016. Web. 28 Oct 2020.

Vancouver:

Se T. Depth and Associated Primes of Modules over a Ring. [Internet] [Doctoral dissertation]. University of Kansas; 2016. [cited 2020 Oct 28]. Available from: http://hdl.handle.net/1808/21895.

Council of Science Editors:

Se T. Depth and Associated Primes of Modules over a Ring. [Doctoral Dissertation]. University of Kansas; 2016. Available from: http://hdl.handle.net/1808/21895