A Study on the Vanishing of Ext

Authors

Andrew J. Soto Levins, University of Nebraska-LincolnFollow

First Advisor

Mark E. Walker

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

Date of this Version

8-2024

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Mark E. Walker

Lincoln, Nebraska, August 2024

Comments

Copyright 2024, Andrew J. Soto Levins. Used by permission

Abstract

This thesis has two goals. The first is to study an Ext analog of the rigidity of Tor, and the second is to study Auslander bounds.

In Chapter 2 we show that if R is an unramified hypersurface, if M and N are finitely generated R-modules, and if the nth Ext modules of M against N is zero for some n less than or equal to the grade of M, then the ith Ext module of M against N is zero for all i less than or equal to n. A corollary of this says that if M is nonzero, then the ith Ext module of M against M is not zero whenever i is greater than or equal to zero and i is less than or equal to the grade of M. We also give an extension of this result to complete intersections. These results are related to a question of Jorgensen and results of Dao.

Chapter 3 is a study on Auslander bounds. The Auslander bound of a module can be thought of as a generalization of projective dimension. We say that the Auslander bound of M is finite if for all finitely generated modules N such that whenever the nth Ext module of M against N is zero for n sufficiently large, there exists an integer b that only depends on M so that the nth Ext module of M against N is zero for n strictly greater than b. In this chapter we give new results on the Auslander bound by weakening the AC condition, generalizing many theorems in the the literature. We then define an Auslander bound for complexes and extend the module results to complexes.

Advisor: Mark E. Walker

Recommended Citation

Soto Levins, Andrew J., "A Study on the Vanishing of Ext" (2024). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–. 134.
https://digitalcommons.unl.edu/dissunl/134