{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71265"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71265","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Projective Resolutions of Generic Order Ideals","abstract":"Let Y$\\sp{\\rm (n)}$ be the 1 x n matrix containing indeterminate entries $\\{$Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\}$ and X$\\sp{\\rm (n)}$ be the n x n alternating matrix containing indeterminate entries $\\{$X$\\sb{\\rm ij}\\vert$1 $\\leq$ i $<$ j $\\leq$ n$\\}$, where we adopt the convention that X$\\sb{\\rm ii}$ = 0 and X$\\sb{\\rm ji}$ = $-$ X$\\sb{\\rm ij}$. Let $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ be the entries of the product Y$\\sp{\\rm (n)}$X$\\sp{\\rm (n)}$. Let R$\\sb{\\rm n}$ be the ring $\\doubz$ (X$\\sb{\\rm ij}$,Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack\\sb{\\rm 1 \\leq i < j \\leq n}$. Then $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n-1}{\\rm (n)}\\}$ is a regular sequence and $\\sum\\sbsp{\\rm i=1}{\\rm n}$ Y$\\sb{\\rm i}$g$\\sbsp{\\rm i}{\\rm (n)}$ = 0. Let I$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ and J$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)},(-1)\\sp{\\rm n+1}$pf(X$\\sp{\\rm (n)})\\}$. Since the pfaffian of X$\\sp{\\rm (2m+1)}$ is zero, J$\\sb{\\rm 2m+1}$ = I$\\sb{\\rm 2m+1}$. I$\\sb{\\rm n}$ is the generic order ideal of the second syzygy M of the Koszul complex resolution of $\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) /(Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) for n $\\geq$ 3, and is the ideal of relations of Sym$\\sb{\\doubz\\lbrack\\rm Y\\sb1,\\dots,Y\\sb{n}\\rbrack}(\\Lambda\\sp{\\rm n-2}(\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack)\\sp{\\rm n}$/M*). Since I$\\sb{\\rm n}$ has grade n $-$ 1 and is generated by n elements, it is an almost complete intersection. Huneke and Ulrich showed that J$\\sb{\\rm n}$ is a perfect prime ideal of grade n $-$ 1 and the ideals J$\\sb{\\rm n+1}$ and (J$\\sb{\\rm n}$,Y$\\sb{\\rm n+1}$) are linked by the regular sequence $\\{$g$\\sbsp{1}{\\rm (n+1)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n+1)}\\}$.","abstract_html":"Let Y$\\sp{\\rm (n)}$ be the 1 x n matrix containing indeterminate entries $\\{$Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\}$ and X$\\sp{\\rm (n)}$ be the n x n alternating matrix containing indeterminate entries $\\{$X$\\sb{\\rm ij}\\vert$1 $\\leq$ i $<$ j $\\leq$ n$\\}$, where we adopt the convention that X$\\sb{\\rm ii}$ = 0 and X$\\sb{\\rm ji}$ = $-$ X$\\sb{\\rm ij}$. Let $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ be the entries of the product Y$\\sp{\\rm (n)}$X$\\sp{\\rm (n)}$. Let R$\\sb{\\rm n}$ be the ring $\\doubz$ (X$\\sb{\\rm ij}$,Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack\\sb{\\rm 1 \\leq i < j \\leq n}$. Then $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n-1}{\\rm (n)}\\}$ is a regular sequence and $\\sum\\sbsp{\\rm i=1}{\\rm n}$ Y$\\sb{\\rm i}$g$\\sbsp{\\rm i}{\\rm (n)}$ = 0. Let I$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ and J$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)},(-1)\\sp{\\rm n+1}$pf(X$\\sp{\\rm (n)})\\}$. Since the pfaffian of X$\\sp{\\rm (2m+1)}$ is zero, J$\\sb{\\rm 2m+1}$ = I$\\sb{\\rm 2m+1}$. I$\\sb{\\rm n}$ is the generic order ideal of the second syzygy M of the Koszul complex resolution of $\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) /(Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) for n $\\geq$ 3, and is the ideal of relations of Sym$\\sb{\\doubz\\lbrack\\rm Y\\sb1,\\dots,Y\\sb{n}\\rbrack}(\\Lambda\\sp{\\rm n-2}(\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack)\\sp{\\rm n}$/M*). Since I$\\sb{\\rm n}$ has grade n $-$ 1 and is generated by n elements, it is an almost complete intersection. Huneke and Ulrich showed that J$\\sb{\\rm n}$ is a perfect prime ideal of grade n $-$ 1 and the ideals J$\\sb{\\rm n+1}$ and (J$\\sb{\\rm n}$,Y$\\sb{\\rm n+1}$) are linked by the regular sequence $\\{$g$\\sbsp{1}{\\rm (n+1)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n+1)}\\}$.","abstract_has_math":true,"creators":["Kim, Saeja Oh"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Grayson, Daniel,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:22Z","date_published":"2014-12-16T06:18:22Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8823170"],"render_values":[{"text":"(UMI)AAI8823170","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71265","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Grayson, Daniel,"]},{"key":"dc:creator","label":"Author","values":["Kim, Saeja Oh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:22Z","10000-01-01","1988"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71265","(UMI)AAI8823170"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let Y$\\sp{\\rm (n)}$ be the 1 x n matrix containing indeterminate entries $\\{$Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\}$ and X$\\sp{\\rm (n)}$ be the n x n alternating matrix containing indeterminate entries $\\{$X$\\sb{\\rm ij}\\vert$1 $\\leq$ i $<$ j $\\leq$ n$\\}$, where we adopt the convention that X$\\sb{\\rm ii}$ = 0 and X$\\sb{\\rm ji}$ = $-$ X$\\sb{\\rm ij}$. Let $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ be the entries of the product Y$\\sp{\\rm (n)}$X$\\sp{\\rm (n)}$. Let R$\\sb{\\rm n}$ be the ring $\\doubz$ (X$\\sb{\\rm ij}$,Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack\\sb{\\rm 1 \\leq i < j \\leq n}$. Then $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n-1}{\\rm (n)}\\}$ is a regular sequence and $\\sum\\sbsp{\\rm i=1}{\\rm n}$ Y$\\sb{\\rm i}$g$\\sbsp{\\rm i}{\\rm (n)}$ = 0. Let I$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ and J$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)},(-1)\\sp{\\rm n+1}$pf(X$\\sp{\\rm (n)})\\}$. Since the pfaffian of X$\\sp{\\rm (2m+1)}$ is zero, J$\\sb{\\rm 2m+1}$ = I$\\sb{\\rm 2m+1}$. I$\\sb{\\rm n}$ is the generic order ideal of the second syzygy M of the Koszul complex resolution of $\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) /(Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) for n $\\geq$ 3, and is the ideal of relations of Sym$\\sb{\\doubz\\lbrack\\rm Y\\sb1,\\dots,Y\\sb{n}\\rbrack}(\\Lambda\\sp{\\rm n-2}(\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack)\\sp{\\rm n}$/M*). Since I$\\sb{\\rm n}$ has grade n $-$ 1 and is generated by n elements, it is an almost complete intersection. Huneke and Ulrich showed that J$\\sb{\\rm n}$ is a perfect prime ideal of grade n $-$ 1 and the ideals J$\\sb{\\rm n+1}$ and (J$\\sb{\\rm n}$,Y$\\sb{\\rm n+1}$) are linked by the regular sequence $\\{$g$\\sbsp{1}{\\rm (n+1)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n+1)}\\}$.","In this thesis, we produce a minimal free resolution of R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$. From this resolution, we read that I$\\sb{\\rm 2n}$ is an almost perfect ideal (i.e. pd(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$) = grade(I$\\sb{\\rm 2n}$) + 1), Ext$\\sbsp{\\rm R\\sb{2n}}{\\rm 2n}$(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$,R$\\sb{\\rm 2n}$) = R$\\sb{\\rm 2n}$/(Y$\\sb1,\\dots$,Y$\\sb{\\rm 2n}$), (Y$\\sb1,\\dots$,Y$\\sb{\\rm 2n}$) $\\in$ Ass(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$) and J$\\sb{\\rm 2n}$ $\\in$ Ass(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$).","Made available in DSpace on 2014-12-16T06:18:22Z (GMT). No. of bitstreams: 1 8823170.pdf: 2372821 bytes, checksum: 48090d09f8f7734eb1f0dd94facd35de (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71431 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","107 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["Projective Resolutions of Generic Order Ideals"]}]}],"canonical_facts":{"dc:contributor":["Grayson, Daniel,"],"dc:creator":["Kim, Saeja Oh"],"dc:date":["2014-12-16T06:18:22Z","10000-01-01","1988"],"dc:description":["Let Y$\\sp{\\rm (n)}$ be the 1 x n matrix containing indeterminate entries $\\{$Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\}$ and X$\\sp{\\rm (n)}$ be the n x n alternating matrix containing indeterminate entries $\\{$X$\\sb{\\rm ij}\\vert$1 $\\leq$ i $<$ j $\\leq$ n$\\}$, where we adopt the convention that X$\\sb{\\rm ii}$ = 0 and X$\\sb{\\rm ji}$ = $-$ X$\\sb{\\rm ij}$. Let $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ be the entries of the product Y$\\sp{\\rm (n)}$X$\\sp{\\rm (n)}$. Let R$\\sb{\\rm n}$ be the ring $\\doubz$ (X$\\sb{\\rm ij}$,Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack\\sb{\\rm 1 \\leq i < j \\leq n}$. Then $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n-1}{\\rm (n)}\\}$ is a regular sequence and $\\sum\\sbsp{\\rm i=1}{\\rm n}$ Y$\\sb{\\rm i}$g$\\sbsp{\\rm i}{\\rm (n)}$ = 0. Let I$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)}\\}$ and J$\\sb{\\rm n}$ be the ideal of R$\\sb{\\rm n}$ generated by $\\{$g$\\sbsp{1}{\\rm (n)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n)},(-1)\\sp{\\rm n+1}$pf(X$\\sp{\\rm (n)})\\}$. Since the pfaffian of X$\\sp{\\rm (2m+1)}$ is zero, J$\\sb{\\rm 2m+1}$ = I$\\sb{\\rm 2m+1}$. I$\\sb{\\rm n}$ is the generic order ideal of the second syzygy M of the Koszul complex resolution of $\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) /(Y$\\sb1,\\dots$,Y$\\sb{\\rm n}$) for n $\\geq$ 3, and is the ideal of relations of Sym$\\sb{\\doubz\\lbrack\\rm Y\\sb1,\\dots,Y\\sb{n}\\rbrack}(\\Lambda\\sp{\\rm n-2}(\\doubz$ (Y$\\sb1,\\dots$,Y$\\sb{\\rm n}\\rbrack)\\sp{\\rm n}$/M*). Since I$\\sb{\\rm n}$ has grade n $-$ 1 and is generated by n elements, it is an almost complete intersection. Huneke and Ulrich showed that J$\\sb{\\rm n}$ is a perfect prime ideal of grade n $-$ 1 and the ideals J$\\sb{\\rm n+1}$ and (J$\\sb{\\rm n}$,Y$\\sb{\\rm n+1}$) are linked by the regular sequence $\\{$g$\\sbsp{1}{\\rm (n+1)},\\dots$,g$\\sbsp{\\rm n}{\\rm (n+1)}\\}$.","In this thesis, we produce a minimal free resolution of R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$. From this resolution, we read that I$\\sb{\\rm 2n}$ is an almost perfect ideal (i.e. pd(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$) = grade(I$\\sb{\\rm 2n}$) + 1), Ext$\\sbsp{\\rm R\\sb{2n}}{\\rm 2n}$(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$,R$\\sb{\\rm 2n}$) = R$\\sb{\\rm 2n}$/(Y$\\sb1,\\dots$,Y$\\sb{\\rm 2n}$), (Y$\\sb1,\\dots$,Y$\\sb{\\rm 2n}$) $\\in$ Ass(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$) and J$\\sb{\\rm 2n}$ $\\in$ Ass(R$\\sb{\\rm 2n}$/I$\\sb{\\rm 2n}$).","Made available in DSpace on 2014-12-16T06:18:22Z (GMT). No. of bitstreams: 1 8823170.pdf: 2372821 bytes, checksum: 48090d09f8f7734eb1f0dd94facd35de (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71431 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","107 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71265","(UMI)AAI8823170"],"dc:subject":["Mathematics"],"dc:title":["Projective Resolutions of Generic Order Ideals"],"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":"2026-07-22T22:26:04Z"}