Betti numbers of Koszul algebras and codimension two matrix factorizations

This thesis consists of two projects on the structure of free resolutions in commutative algebra. After developing some necessary background, we prove a structure theorem in Chapter 3 for the defining ideals of Koszul almost complete intersections and, in the process, give an affirmative answer for all such rings to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras. In Chapter 4, we study the codimension two matrix factorizations of Eisenbud and Peeva. Each matrix factorization compactly encodes the data of a free resolution of its corresponding matrix factorization module. By showing that each matrix factorization also encodes a canonical system of higher homotopies on this free resolution, we are able to construct a functor from codimension two matrix factorizations to the singularity category of the corresponding complete intersection. This represents the first step towards reconciling higher codimension matrix factorizations with known generalizations of a theorem of Buchweitz and Orlov in the hypersurface case.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Mastroeni, Matthew N

Contributors dc:contributor

• Schenck, Hal
• Katz, Sheldon
• Dutta, Sankar
• Griffith, Phil

• Koszul algebras
• almost complete intersections
• Betti numbers
• free resolutions
• matrix factorizations

Statement dc:rights

• Copyright 2018 Matthew Mastroeni

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101658

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101658