{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/42274"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/42274","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Grade Conjecture and asymptotic intersection multiplicity","abstract":"In this thesis, we study Peskine and Szpiro's Grade Conjecture and its connection with asymptotic intersection multiplicity $\\chi_\\infty$. Given an $A$-module $M$ of finite projective dimension and a system of parameters $x_1, \\ldots, x_r$ for $M$, we show, under certain assumptions on $M$, that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also give a necessary and sufficient condition on $M$ for the existence of a system of parameters $\\underline{x}$ with $\\chi_\\infty(M, A/\\underline{x}) > 0$. We then prove that if the Grade Conjecture holds for a given module $M$, then there is a system of parameters $\\underline{x}$ such that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.","abstract_html":"In this thesis, we study Peskine and Szpiro&#x27;s Grade Conjecture and its connection with asymptotic intersection multiplicity <span class=\"etd-inline-math\">\\chi<sub>\\</sub>infty</span>. Given an $A$-module $M$ of finite projective dimension and a system of parameters <span class=\"etd-inline-math\">x<sub>1</sub>, \\ldots, x<sub>r</sub></span> for $M$, we show, under certain assumptions on $M$, that <span class=\"etd-inline-math\">\\chi<sub>\\</sub>infty(M, A/\\underline{x}) &gt; 0</span>. We also give a necessary and sufficient condition on $M$ for the existence of a system of parameters $\\underline{x}$ with <span class=\"etd-inline-math\">\\chi<sub>\\</sub>infty(M, A/\\underline{x}) &gt; 0</span>. We then prove that if the Grade Conjecture holds for a given module $M$, then there is a system of parameters $\\underline{x}$ such that <span class=\"etd-inline-math\">\\chi<sub>\\</sub>infty(M, A/\\underline{x}) &gt; 0</span>. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.","abstract_has_math":true,"creators":["Beder, Jesse"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dutta, Sankar P.","Griffith, Phillip A.","Schenck, Henry K.","Haboush, William J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-02-03T19:29:58Z","date_published":"2013-02-03T19:29:58Z","updated_at":"2026-07-22T22:25:33Z","subjects":["commutative algebra","grade conjecture","characteristic p","frobenius","intersection multiplicity"],"languages":["en"],"rights":["Copyright 2012 Jesse Beder"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/42274","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dutta, Sankar P.","Griffith, Phillip A.","Schenck, Henry K.","Haboush, William J."]},{"key":"dc:creator","label":"Author","values":["Beder, Jesse"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-02-03T19:29:58Z","2012-12"]},{"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":["commutative algebra","grade conjecture","characteristic p","frobenius","intersection multiplicity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Jesse Beder"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/42274"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study Peskine and Szpiro's Grade Conjecture and its connection with asymptotic intersection multiplicity $\\chi_\\infty$. Given an $A$-module $M$ of finite projective dimension and a system of parameters $x_1, \\ldots, x_r$ for $M$, we show, under certain assumptions on $M$, that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also give a necessary and sufficient condition on $M$ for the existence of a system of parameters $\\underline{x}$ with $\\chi_\\infty(M, A/\\underline{x}) > 0$. We then prove that if the Grade Conjecture holds for a given module $M$, then there is a system of parameters $\\underline{x}$ such that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-18T12:47:30Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 7 uiucthesis.cls: 17745 bytes, checksum: 2d97b89eb400b38a73a704ef757b5460 (MD5) 4-main.tex: 13848 bytes, checksum: fe7bd516281d0223ea8dbba70594ee4a (MD5) 3-grade-conjecture.tex: 42579 bytes, checksum: 78fc7491eedac22dd5c29aa607e86f24 (MD5) 2-background.tex: 28120 bytes, checksum: 0f5e758e7192bb82a169f4a12dfad857 (MD5) 1-introduction.tex: 7401 bytes, checksum: 2843074c02d5facfcd9ec77f66b995ef (MD5) thesis.tex: 5772 bytes, checksum: 2d8e71e41a98a106ae13e84347eb3c0a (MD5) Beder_Jesse.pdf: 390679 bytes, checksum: b9c2a0000d711c1da1c5e19e390fe989 (MD5)","Made available in DSpace on 2013-02-03T19:29:58Z (GMT). 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Given an $A$-module $M$ of finite projective dimension and a system of parameters $x_1, \\ldots, x_r$ for $M$, we show, under certain assumptions on $M$, that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also give a necessary and sufficient condition on $M$ for the existence of a system of parameters $\\underline{x}$ with $\\chi_\\infty(M, A/\\underline{x}) > 0$. We then prove that if the Grade Conjecture holds for a given module $M$, then there is a system of parameters $\\underline{x}$ such that $\\chi_\\infty(M, A/\\underline{x}) > 0$. We also prove the Grade Conjecture for complete equidimensional local rings in any characteristic.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-18T12:47:30Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 7 uiucthesis.cls: 17745 bytes, checksum: 2d97b89eb400b38a73a704ef757b5460 (MD5) 4-main.tex: 13848 bytes, checksum: fe7bd516281d0223ea8dbba70594ee4a (MD5) 3-grade-conjecture.tex: 42579 bytes, checksum: 78fc7491eedac22dd5c29aa607e86f24 (MD5) 2-background.tex: 28120 bytes, checksum: 0f5e758e7192bb82a169f4a12dfad857 (MD5) 1-introduction.tex: 7401 bytes, checksum: 2843074c02d5facfcd9ec77f66b995ef (MD5) thesis.tex: 5772 bytes, checksum: 2d8e71e41a98a106ae13e84347eb3c0a (MD5) Beder_Jesse.pdf: 390679 bytes, checksum: b9c2a0000d711c1da1c5e19e390fe989 (MD5)","Made available in DSpace on 2013-02-03T19:29:58Z (GMT). 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