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University of Kansas

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

Degree: PhD, Mathematics, 2016, University of Kansas

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/In M) and the values \operatorname{depth}J F(M/In M) become independent of n for large n. In the second topic, we consider rings of the form R = k[xa,xp1yq1, …,xptyqt,yb], 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

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

APA (6th 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 (16th 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 (7th 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

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