{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86980"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86980","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A Variation on a Theme of Vasconcelos","abstract":"In 1967, Vasconcelos and Ferrand independently demonstrated that an ideal I in a Noetherian local ring R is generated by a regular sequence if and only if I/ I2 is free over R/I and pdRI < infinity. In this thesis, we prove that a radical ideal I in an excellent local normal domain is generated by a regular sequence provided R/I satisfies the Serre condition S2, I/ I(2) is R/I-free, and I is generated by a regular sequence when localized at primes P containing I such that hat P/I &le; 1. In particular, we are able to replace the strong assumption pdRI < infinity with essentially weaker hypotheses. Our proof makes use of lifting as presented by Auslander, Ding and Slobbered [1993] together with a finer analysis of the elimination of the obstructions to lifting cyclic modules. Subsequently, we develop some criteria for lifting a module M in a more general setting. Our approach here is to again assume the conclusion holds when localizing at primes of low codimension while also requiring M and a related module of morphemes to possess sufficient depth in the remaining cases.","abstract_html":"In 1967, Vasconcelos and Ferrand independently demonstrated that an ideal I in a Noetherian local ring R is generated by a regular sequence if and only if I/ I2 is free over R/I and pdRI &lt; infinity. In this thesis, we prove that a radical ideal I in an excellent local normal domain is generated by a regular sequence provided R/I satisfies the Serre condition S2, I/ I(2) is R/I-free, and I is generated by a regular sequence when localized at primes P containing I such that hat P/I &amp;le; 1. In particular, we are able to replace the strong assumption pdRI &lt; infinity with essentially weaker hypotheses. Our proof makes use of lifting as presented by Auslander, Ding and Slobbered [1993] together with a finer analysis of the elimination of the obstructions to lifting cyclic modules. Subsequently, we develop some criteria for lifting a module M in a more general setting. Our approach here is to again assume the conclusion holds when localizing at primes of low codimension while also requiring M and a related module of morphemes to possess sufficient depth in the remaining cases.","abstract_has_math":false,"creators":["Smith, Daniel Aaron"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Griffith, Phillip A."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:27Z","date_published":"2015-09-28T15:20:27Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9945003"],"render_values":[{"text":"(MiAaPQ)AAI9945003","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86980","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Griffith, Phillip A."]},{"key":"dc:creator","label":"Author","values":["Smith, Daniel Aaron"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:27Z","10000-01-01","1999"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86980","(MiAaPQ)AAI9945003"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In 1967, Vasconcelos and Ferrand independently demonstrated that an ideal I in a Noetherian local ring R is generated by a regular sequence if and only if I/ I2 is free over R/I and pdRI < infinity. In this thesis, we prove that a radical ideal I in an excellent local normal domain is generated by a regular sequence provided R/I satisfies the Serre condition S2, I/ I(2) is R/I-free, and I is generated by a regular sequence when localized at primes P containing I such that hat P/I &le; 1. In particular, we are able to replace the strong assumption pdRI < infinity with essentially weaker hypotheses. Our proof makes use of lifting as presented by Auslander, Ding and Slobbered [1993] together with a finer analysis of the elimination of the obstructions to lifting cyclic modules. Subsequently, we develop some criteria for lifting a module M in a more general setting. Our approach here is to again assume the conclusion holds when localizing at primes of low codimension while also requiring M and a related module of morphemes to possess sufficient depth in the remaining cases.","Made available in DSpace on 2015-09-28T15:20:27Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9945003.pdf: 1183412 bytes, checksum: e0aee14a72465fd3b53484117ccc013f (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88261 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","24 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."]},{"key":"dc:title","label":"Title","values":["A Variation on a Theme of Vasconcelos"]}]}],"canonical_facts":{"dc:contributor":["Griffith, Phillip A."],"dc:creator":["Smith, Daniel Aaron"],"dc:date":["2015-09-28T15:20:27Z","10000-01-01","1999"],"dc:description":["In 1967, Vasconcelos and Ferrand independently demonstrated that an ideal I in a Noetherian local ring R is generated by a regular sequence if and only if I/ I2 is free over R/I and pdRI < infinity. In this thesis, we prove that a radical ideal I in an excellent local normal domain is generated by a regular sequence provided R/I satisfies the Serre condition S2, I/ I(2) is R/I-free, and I is generated by a regular sequence when localized at primes P containing I such that hat P/I &le; 1. In particular, we are able to replace the strong assumption pdRI < infinity with essentially weaker hypotheses. Our proof makes use of lifting as presented by Auslander, Ding and Slobbered [1993] together with a finer analysis of the elimination of the obstructions to lifting cyclic modules. Subsequently, we develop some criteria for lifting a module M in a more general setting. Our approach here is to again assume the conclusion holds when localizing at primes of low codimension while also requiring M and a related module of morphemes to possess sufficient depth in the remaining cases.","Made available in DSpace on 2015-09-28T15:20:27Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9945003.pdf: 1183412 bytes, checksum: e0aee14a72465fd3b53484117ccc013f (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88261 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","24 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."],"dc:identifier":["http://hdl.handle.net/2142/86980","(MiAaPQ)AAI9945003"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["A Variation on a Theme of Vasconcelos"],"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":"2026-07-22T22:26:28Z"}