Betti numbers of Koszul algebras and codimension two matrix factorizations

Mastroeni, Matthew N

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https://hdl.handle.net/2142/101658 Copy

Description

Title

Betti numbers of Koszul algebras and codimension two matrix factorizations

Author(s)

Mastroeni, Matthew N

Issue Date

2018-06-28

Director of Research (if dissertation) or Advisor (if thesis)

Schenck, Hal

Doctoral Committee Chair(s)

Katz, Sheldon

Committee Member(s)

• Dutta, Sankar
• Griffith, Phil

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2018-08

Type of Resource

text

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

Copyright and License Information

Copyright 2018 Matthew Mastroeni

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