{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/82612"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/82612","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Graded artin algebras, coverings and factor rings","abstract":"Let (Γ,ρ) be a directed graph with relations. Let F: Γ’ → Γ be a topological covering. It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/<ρ̅>-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/<ρ>-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/<ρ>-modules does not depend on <ρ>. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/<ρ>-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/<ρ> is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules.","abstract_html":"Let (Γ,ρ) be a directed graph with relations. Let F: Γ’ → Γ be a topological covering. It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/&lt;ρ̅&gt;-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/&lt;ρ&gt;-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/&lt;ρ&gt;-modules does not depend on &lt;ρ&gt;. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/&lt;ρ&gt;-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/&lt;ρ&gt; is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules.","abstract_has_math":false,"creators":["Weaver, Martha Ellen"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Green, Edward"],"committee_members":["Herdman, Terry L.","Fletcher, Peter","Brown, E.A.","Arnold, Jesse T."],"year":1986,"date_issued":"1986","date_published":"1986","updated_at":"2026-07-22T22:19:41Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/82612","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"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Herdman, Terry L.","Fletcher, Peter","Brown, E.A.","Arnold, Jesse T."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Weaver, Martha Ellen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-03-26T18:50:30Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-03-26T18:50:30Z"]},{"key":"dc:date.issued","label":"Date","values":["1986"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"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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It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/<ρ̅>-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/<ρ>-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/<ρ>-modules does not depend on <ρ>. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/<ρ>-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/<ρ> is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Graded artin algebras, coverings and factor rings"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Green, Edward"],"dc:contributor.committeemember":["Herdman, Terry L.","Fletcher, Peter","Brown, E.A.","Arnold, Jesse T."],"dc:contributor.department":["Mathematics"],"dc:creator":["Weaver, Martha Ellen"],"dc:date.accessioned":["2018-03-26T18:50:30Z"],"dc:date.available":["2018-03-26T18:50:30Z"],"dc:date.issued":["1986"],"dc:description.abstract":["Let (Γ,ρ) be a directed graph with relations. Let F: Γ’ → Γ be a topological covering. It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/<ρ̅>-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/<ρ>-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/<ρ>-modules does not depend on <ρ>. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/<ρ>-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/<ρ> is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/82612"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Graded artin algebras, coverings and factor rings"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:41Z"}