{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/37649"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/37649","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Syzygy Decompositions and Projective Resolutions","abstract":"We give a projective resolution of a finite dimensional π«-algebra π¬ over its enveloping algebra π¬π = π¬ππ β¨π«π¬. The description of this resolution is related to decompositions of the first syzygy module of π¬ as an π¬π module. Resolutions of right π¬ modules ππ¬ may be obtained by tensoring π over π¬ with this bimodule resoution. We describe how to obtain such a resolution when π is simple or when π is given in the form of a projective presentation. Computations of πΈπ₯π‘π¬π(ππ£,ππ€) for certain classes of algebras π¬ are made using these resolutions, and applied to obtain results on global dimension.","abstract_html":"We give a projective resolution of a finite dimensional π«-algebra π¬ over its enveloping algebra π¬<SUP>π</SUP> = π¬<SUP>ππ</SUP> β¨<SUB>π«</SUB>π¬. The description of this resolution is related to decompositions of the first syzygy module of π¬ as an π¬<SUP>π</SUP> module. Resolutions of right π¬ modules π<SUB>π¬</SUB> may be obtained by tensoring π over π¬ with this bimodule resoution. We describe how to obtain such a resolution when π is simple or when π is given in the form of a projective presentation. Computations of <I>πΈπ₯π‘</I><SUB>π¬</SUB><SUP>π</SUP>(π<SUB>π£</SUB>,π<SUB>π€</SUB>) for certain classes of algebras π¬ are made using these resolutions, and applied to obtain results on global dimension.","abstract_has_math":false,"creators":["Smith, Nathan A."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Green, Edward L."],"committee_members":["Haskell, Peter E.","Linnell, Peter A.","Rossi, John F.","Thomson, James E."],"year":1999,"date_issued":"1999-04-16","date_published":"1999-04-16","updated_at":"2026-07-22T22:20:41Z","subjects":["Syzygy","Resolution","Algebra","Module","Ring","Decomposition"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-042199-120414"],"render_values":[{"text":"etd-042199-120414","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/37649","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Green, Edward L."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Haskell, Peter E.","Linnell, Peter A.","Rossi, John F.","Thomson, James E."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Smith, Nathan A."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:10:34Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:10:34Z","2000-04-24"]},{"key":"dc:date.issued","label":"Date","values":["1999-04-16"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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The description of this resolution is related to decompositions of the first syzygy module of π¬ as an π¬π module. Resolutions of right π¬ modules ππ¬ may be obtained by tensoring π over π¬ with this bimodule resoution. We describe how to obtain such a resolution when π is simple or when π is given in the form of a projective presentation. Computations of πΈπ₯π‘π¬π(ππ£,ππ€) for certain classes of algebras π¬ are made using these resolutions, and applied to obtain results on global dimension."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Syzygy Decompositions and Projective Resolutions"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Green, Edward L."],"dc:contributor.committeemember":["Haskell, Peter E.","Linnell, Peter A.","Rossi, John F.","Thomson, James E."],"dc:contributor.department":["Mathematics"],"dc:creator":["Smith, Nathan A."],"dc:date.accessioned":["2014-03-14T21:10:34Z"],"dc:date.available":["2014-03-14T21:10:34Z","2000-04-24"],"dc:date.issued":["1999-04-16"],"dc:description.abstract":["We give a projective resolution of a finite dimensional π«-algebra π¬ over its enveloping algebra π¬π = π¬ππ β¨π«π¬. The description of this resolution is related to decompositions of the first syzygy module of π¬ as an π¬π module. Resolutions of right π¬ modules ππ¬ may be obtained by tensoring π over π¬ with this bimodule resoution. We describe how to obtain such a resolution when π is simple or when π is given in the form of a projective presentation. Computations of πΈπ₯π‘π¬π(ππ£,ππ€) for certain classes of algebras π¬ are made using these resolutions, and applied to obtain results on global dimension."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-042199-120414"],"dc:identifier.uri":["http://hdl.handle.net/10919/37649"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Syzygy","Resolution","Algebra","Module","Ring","Decomposition"],"dc:title":["Syzygy Decompositions and Projective Resolutions"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:41Z"}