{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86931"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86931","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On the Strong Direct Summand Conjecture","abstract":"In this thesis, our aim is the study the Vanishing of Maps of Tor Conjecture of Hochster and Huneke. We mainly focus on an equivalent characterization called the Strong Direct Summand Conjecture, due to N. Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in mixed characteristic using knowledge about splittings in lower dimensions. In particular, we show that the vanishing of the first Ext module allows us to lift a splitting in lower dimension and prove the Strong Direct Summand Conjecture. In Chapter 4, we study the related Strong Monomial Conjecture. Extending work of Dutta, we give several reformulations of the Strong Monomial Conjecture and prove the Strong Monomial Conjecture for systems of parameters of a certain form. In Chapter 5, we present a generalization of the Vanishing Maps of Tor Conjecture and prove the equicharacteristic case. We then give a shorter proof of the Strong Direct Summand Conjecture.","abstract_html":"In this thesis, our aim is the study the Vanishing of Maps of Tor Conjecture of Hochster and Huneke. We mainly focus on an equivalent characterization called the Strong Direct Summand Conjecture, due to N. Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in mixed characteristic using knowledge about splittings in lower dimensions. In particular, we show that the vanishing of the first Ext module allows us to lift a splitting in lower dimension and prove the Strong Direct Summand Conjecture. In Chapter 4, we study the related Strong Monomial Conjecture. Extending work of Dutta, we give several reformulations of the Strong Monomial Conjecture and prove the Strong Monomial Conjecture for systems of parameters of a certain form. In Chapter 5, we present a generalization of the Vanishing Maps of Tor Conjecture and prove the equicharacteristic case. We then give a shorter proof of the Strong Direct Summand Conjecture.","abstract_has_math":false,"creators":["McCullough, Jason"],"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":"2015-09-28T15:20:13Z","date_published":"2015-09-28T15:20:13Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Theoretical Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3392213"],"render_values":[{"text":"(MiAaPQ)AAI3392213","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86931","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":["McCullough, Jason"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:13Z","10000-01-01","2009"]},{"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":["Theoretical 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/2142/86931","(MiAaPQ)AAI3392213"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, our aim is the study the Vanishing of Maps of Tor Conjecture of Hochster and Huneke. We mainly focus on an equivalent characterization called the Strong Direct Summand Conjecture, due to N. Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in mixed characteristic using knowledge about splittings in lower dimensions. In particular, we show that the vanishing of the first Ext module allows us to lift a splitting in lower dimension and prove the Strong Direct Summand Conjecture. In Chapter 4, we study the related Strong Monomial Conjecture. Extending work of Dutta, we give several reformulations of the Strong Monomial Conjecture and prove the Strong Monomial Conjecture for systems of parameters of a certain form. In Chapter 5, we present a generalization of the Vanishing Maps of Tor Conjecture and prove the equicharacteristic case. We then give a shorter proof of the Strong Direct Summand Conjecture.","Made available in DSpace on 2015-09-28T15:20:13Z (GMT). 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Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in mixed characteristic using knowledge about splittings in lower dimensions. In particular, we show that the vanishing of the first Ext module allows us to lift a splitting in lower dimension and prove the Strong Direct Summand Conjecture. In Chapter 4, we study the related Strong Monomial Conjecture. Extending work of Dutta, we give several reformulations of the Strong Monomial Conjecture and prove the Strong Monomial Conjecture for systems of parameters of a certain form. In Chapter 5, we present a generalization of the Vanishing Maps of Tor Conjecture and prove the equicharacteristic case. We then give a shorter proof of the Strong Direct Summand Conjecture.","Made available in DSpace on 2015-09-28T15:20:13Z (GMT). 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