## Mathematics, Department of

#### First Advisor

Luchezar L. Avramov

#### Second Advisor

Mark E. Walker

#### Date of this Version

Summer 8-2019

#### Abstract

Let *R* be a commutative noetherian ring. A well-known theorem in commutative algebra states that *R* is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known ``localization problem" for regular local rings. The main result of this thesis is establishing an analogous characterization for when *R* is locally a complete intersection. Namely, *R* is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a nontrivial perfect complex in the derived category using finitely many cones and retracts. This answers a question of Dwyer, Greenlees and Iyengar posed in 2006 and yields a completely triangulated category characterization of locally complete intersection rings. Moreover, this work gives a new proof that a complete intersection localizes.

Advisors: Luchezar L. Avramov and Mark E. Walker

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Luchezar L. Avramov and Mark E. Walker. Lincoln, Nebraska/; August, 2019

Copyright 2019 Joshua Pollitz