{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19766"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19766","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On lower bounds for the betti numbers of finite length modules","abstract":"In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded.","abstract_html":"In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded.","abstract_has_math":false,"creators":["Charalambous, Hara"],"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":2011,"date_issued":"2011-05-07T12:17:51Z","date_published":"2011-05-07T12:17:51Z","updated_at":"2026-07-22T22:25:14Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Charalambous, Hara"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9026154","AAI9026154"],"render_values":[{"text":"(UMI)AAI9026154","href":null,"code":true},{"text":"AAI9026154","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19766","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":["Charalambous, Hara"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:17:51Z","10000-01-01","1990"]},{"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"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Charalambous, Hara"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9026154","http://hdl.handle.net/2142/19766","AAI9026154"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded.","In Chapter 2 we show that the betti numbers of multigraded modules satisfy the bound: $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d})$. The multigraded modules that are not isomorphic to R modulo an R-sequence can be divided into two categories according to their betti numbers. Either for all i $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d}) + (\\sbsp{i-1}{d-1})$ or for all i $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d}) + (\\sbsp{\\ i}{d-1})$. Thus if $\\beta\\sbsp{i}{R}(M)$ = $(\\sbsp{i}{d})$ then M must be R modulo a maximal R-sequence. In case M is an almost complete intersection we derive its betti numbers.","In Chapter 3 we give better bounds for the betti numbers of cyclic multigraded modules using a deformation argument combined with localization. Lastly we point out that the sum of the betti numbers for all modules of finite length is greater than or equal to 2$\\sp{d}$ + 2$\\sp{d-1}$ where d is the dimension of the ring up to dimension 4.","Made available in DSpace on 2011-05-07T12:17:51Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9026154.pdf: 1903811 bytes, checksum: ac17ec844ea5bb9444970f23bb333c61 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:39:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:30-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["On lower bounds for the betti numbers of finite length modules"]}]}],"canonical_facts":{"dc:contributor":["Griffith, Phillip A."],"dc:creator":["Charalambous, Hara"],"dc:date":["2011-05-07T12:17:51Z","10000-01-01","1990"],"dc:description":["In this manuscript we consider multigraded modules. Chapter 1 gives the necessary definitions and examples that develop the theory of multigraded modules. A multigraded module has a multigraded minimal resolution. We give the necessary conditions for a matrix to correspond to a multigraded map and show that all associated primes of a multigraded module are multigraded.","In Chapter 2 we show that the betti numbers of multigraded modules satisfy the bound: $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d})$. The multigraded modules that are not isomorphic to R modulo an R-sequence can be divided into two categories according to their betti numbers. Either for all i $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d}) + (\\sbsp{i-1}{d-1})$ or for all i $\\beta\\sbsp{i}{R}(M) \\geq (\\sbsp{i}{d}) + (\\sbsp{\\ i}{d-1})$. Thus if $\\beta\\sbsp{i}{R}(M)$ = $(\\sbsp{i}{d})$ then M must be R modulo a maximal R-sequence. In case M is an almost complete intersection we derive its betti numbers.","In Chapter 3 we give better bounds for the betti numbers of cyclic multigraded modules using a deformation argument combined with localization. Lastly we point out that the sum of the betti numbers for all modules of finite length is greater than or equal to 2$\\sp{d}$ + 2$\\sp{d-1}$ where d is the dimension of the ring up to dimension 4.","Made available in DSpace on 2011-05-07T12:17:51Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9026154.pdf: 1903811 bytes, checksum: ac17ec844ea5bb9444970f23bb333c61 (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:39:17Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:30-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["(UMI)AAI9026154","http://hdl.handle.net/2142/19766","AAI9026154"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Charalambous, Hara"],"dc:subject":["Mathematics"],"dc:title":["On lower bounds for the betti numbers of finite length modules"],"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":"2026-07-22T22:25:14Z"}