{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101658"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101658","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Betti numbers of Koszul algebras and codimension two 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.","abstract_html":"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.","abstract_has_math":false,"creators":["Mastroeni, Matthew N"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Schenck, Hal","Katz, Sheldon","Dutta, Sankar","Griffith, Phil"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-27T16:30:14Z","date_published":"2018-09-27T16:30:14Z","updated_at":"2026-07-22T22:24:40Z","subjects":["Koszul algebras","almost complete intersections","Betti numbers","free resolutions","matrix factorizations"],"languages":["en"],"rights":["Copyright 2018 Matthew Mastroeni"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101658","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schenck, Hal","Katz, Sheldon","Dutta, Sankar","Griffith, Phil"]},{"key":"dc:creator","label":"Author","values":["Mastroeni, Matthew N"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-27T16:30:14Z","2020-09-28T09:15:13Z","2018-06-28","2018-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Koszul algebras","almost complete intersections","Betti numbers","free resolutions","matrix factorizations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Matthew Mastroeni"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101658"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-08-01","The student, Matthew Mastroeni, accepted the attached license on 2018-06-27 at 12:18.","The student, Matthew Mastroeni, submitted this Dissertation for approval on 2018-06-27 at 12:33.","This Dissertation was approved for publication on 2018-06-28 at 11:56.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12676 on 2018-09-27 at 11:16:09","Made available in DSpace on 2018-09-27T16:30:14Z (GMT). No. of bitstreams: 2 MASTROENI-DISSERTATION-2018.pdf: 478509 bytes, checksum: 25720cb3828353404aba12968907f25d (MD5) LICENSE.txt: 4214 bytes, checksum: 589255fccd66caf76c464d80d44845bc (MD5) Previous issue date: 2018-06-28","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:30:34Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:31:43Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:34:29Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107756 on 2020-09-28T09:15:13Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Betti numbers of Koszul algebras and codimension two matrix factorizations"]}]}],"canonical_facts":{"dc:contributor":["Schenck, Hal","Katz, Sheldon","Dutta, Sankar","Griffith, Phil"],"dc:creator":["Mastroeni, Matthew N"],"dc:date":["2018-09-27T16:30:14Z","2020-09-28T09:15:13Z","2018-06-28","2018-08"],"dc:description":["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.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2020-08-01","The student, Matthew Mastroeni, accepted the attached license on 2018-06-27 at 12:18.","The student, Matthew Mastroeni, submitted this Dissertation for approval on 2018-06-27 at 12:33.","This Dissertation was approved for publication on 2018-06-28 at 11:56.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12676 on 2018-09-27 at 11:16:09","Made available in DSpace on 2018-09-27T16:30:14Z (GMT). No. of bitstreams: 2 MASTROENI-DISSERTATION-2018.pdf: 478509 bytes, checksum: 25720cb3828353404aba12968907f25d (MD5) LICENSE.txt: 4214 bytes, checksum: 589255fccd66caf76c464d80d44845bc (MD5) Previous issue date: 2018-06-28","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:30:34Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:31:43Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 107756 Lift date: 2020-09-27T16:34:29Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 107756 on 2020-09-28T09:15:13Z."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/101658"],"dc:language":["en"],"dc:rights":["Copyright 2018 Matthew Mastroeni"],"dc:subject":["Koszul algebras","almost complete intersections","Betti numbers","free resolutions","matrix factorizations"],"dc:title":["Betti numbers of Koszul algebras and codimension two matrix factorizations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:40Z"}