Date of Award

May 2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Claudia Miller

Keywords

complete resolutions, edge ideals, free resolutions, J-closed modules, Koszul homology

Subject Categories

Physical Sciences and Mathematics

Abstract

The Koszul homology algebra of a commutative local (or graded) ring R tends to reflect important information about the ring R and its properties. We examine Koszul homology and the relationship between Koszul homology and minimal free resolutions in various settings. We provide tools which allow one to view Koszul homology in a concrete way, and use these tools in a number of applications.

One setting in which we examine Koszul homology is provided by the notion of J-closed modules. We introduce this new class of modules and present explicit formulas for the generators of Koszul homology with coefficients in such modules. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. We use these formulas to study connections between J-closed ideals and the ideals by which they were inspired, namely, weak complete intersection ideals.

As another application of such formulas, we study the Koszul homology algebra of quotients by edge ideals. We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides an answer, for such

rings, to a question of Avramov about the Koszul homology algebra of a Koszul algebra. In order to obtain this result, we explicitly construct the minimal graded free resolution of the quotient by an edge ideal of a tree.

We also use these formulas to construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. The existence of such a complete resolution gives an isomorphism between certain stable homology and cohomology modules.

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Open Access

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