Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules

TL;DR

This dissertation investigates the long-term behavior of prime divisors, complexities, and regularity in algebraic structures like complete intersection rings, and characterizes regular local rings through syzygy modules.

Contribution

It provides new insights into asymptotic properties of prime divisors, complexities, and regularity, and offers characterizations of regular local rings using syzygy modules.

Findings

Asymptotic stability of complexities established

Linear bounds for Castelnuovo-Mumford regularity derived

Characterizations of regular local rings via syzygy modules

Abstract

This dissertation focuses on the following topics: (1) asymptotic prime divisors over complete intersection rings, (2) asymptotic stability of complexities over complete intersection rings, (3) asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules, and (4) characterizations of regular local rings via syzygy modules of the residue field.