Semester

Fall

Date of Graduation

2023

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Olgur Celikbas

Committee Member

Ela Celikbas

Committee Member

Hong-Jian Lai

Committee Member

Charis Tsikkou

Committee Member

Yongwei Yao

Abstract

This dissertation represents an in-depth exploration of two distinct yet interconnected research topics within commutative algebra: one centered around a conjecture of Huneke and R. Wiegand and the other concerns a depth inequality of Auslander. It consists of the following three papers as well as the author's work under the direction of Professor Olgur Celikbas:

  • Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology, Journal of Mathematical Society of Japan Advance Publication. (joint work with Olgur Celikbas, Hiroki Matsui, and Arash Sadeghi).
  • An extension of a depth inequality of Auslander, Taiwanese Journal of Mathematics, volume 1, no. 1, pages 1-24 (2022). (joint work with Olgur Celikbas and Hiroki Matsui).
  • On the depth and reflexivity of tensor products, Journal of Algebra, volume 606, pages 916-932 (2022). (joint work with Olgur Celikbas and Hiroki Matsui).

The first research topic of this thesis centers around the Huneke-Wiegand Conjecture, which has been a long-standing problem. Building upon the foundational work of Huneke and R. Wiegand, and O. Celikbas, we provide affirmative results for this conjecture in specific cases. In particular, we establish results for 2-periodic modules over local domains. Following the publication of the first paper, the author extended the affirmative result for the conjecture of Huneke and Wiegand to 2-periodic modules over local rings, as well as 4-periodic modules over Gorenstein rings. This thesis also provides a condition for which the conjecture holds for any periodic module. Our goal is to offer a comprehensive understanding of the Huneke-Wiegand Conjecture and potentially uncover new insights into the nature of this conjecture.

The second research topic focuses on a characteristic property of Tor-rigid modules, that is, each regular sequence of a module is also a regular sequence of its underlying ring. This characteristic is equivalent to a depth inequality, which is known as the Auslander's Inequality. By thoroughly examining this inequality, we proceed to extend it in two distinct forms: a natural extension and a syzygy extension. These extensions allow us to explore the conditions under which the extended inequalities hold, thereby providing new insights into the Auslander's Inequality and its far-reaching implications. Moreover, this thesis goes beyond the theoretical exploration. It includes a practical application of the extended inequalities. By employing these extended inequalities, we are able to address a specific problem related to the Auslander Inequality, presenting an innovative solution and further highlighting the practical significance of our research findings.

Download

Included in

Algebra Commons

Share

COinS