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Title:Betti numbers of Koszul algebras and codimension two matrix factorizations
Author(s):Mastroeni, Matthew N
Director of Research:Schenck, Hal
Doctoral Committee Chair(s):Katz, Sheldon
Doctoral Committee Member(s):Dutta, Sankar; Griffith, Phil
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Koszul algebras
almost complete intersections
Betti numbers
free resolutions
matrix factorizations
Abstract: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.
Issue Date:2018-06-28
Rights Information:Copyright 2018 Matthew Mastroeni
Date Available in IDEALS:2018-09-27
Date Deposited:2018-08

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