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