{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/39431"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/39431","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Resolutions mod I, Golod pairs","abstract":"Let R be a commutative ring, I be an ideal in R and let M be a R/ I -module. In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then ∐∞n=o TorR/In (M, N) is a graded module over the polynomial ring R/ I [X₁. .. Xr] with deg Xi = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (ΩnR/IR/I(M),I) is a Golod pair over R where ΩnR/IR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ω¹R/IR/I(M),I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R. Finally we prove that if (M, I) is a Golod pair over R and if a ∈ I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π : R → R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps π₁ : R → R/(a) and π₂ : R/(a) → R/1 are Golod homomorphisms.","abstract_html":"Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext<sub>R/I</sub><i>(M,N)</i> -> Ext<sub>R</sub><i>(M,N)</i> and Tor<sup>R</sup><sub>N</sub><i>(M, N)</i> -> Tor<sup>R/I</sup><sub>n</sub><i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞<sub>n=o</sub> Tor<sup>R/I</sup><sub>n</sub> <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X<sub>r</sub>] with deg X<sub>i</sub> = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ω<sup>n</sup><sub>R/I</sub>R/I(M),I) is a Golod pair over <i>R</i> where Ω<sup>n</sup><sub>R/I</sub>R/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹<sub>R/I</sub>R/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>. Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms.","abstract_has_math":false,"creators":["Gokhale, Dhananjay R."],"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":["Farkas, Daniel R.","Thomson, James E.","McCoy, Robert A.","Arnold, Jimmy T."],"year":1992,"date_issued":"1992-04-05","date_published":"1992-04-05","updated_at":"2026-07-22T22:19:12Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-09202005-091014"],"render_values":[{"text":"etd-09202005-091014","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/39431","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":["Farkas, Daniel R.","Thomson, James E.","McCoy, Robert A.","Arnold, Jimmy T."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Gokhale, Dhananjay R."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:19:09Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:19:09Z","2005-09-20"]},{"key":"dc:date.issued","label":"Date","values":["1992-04-05"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"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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In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then ∐∞n=o TorR/In (M, N) is a graded module over the polynomial ring R/ I [X₁. .. Xr] with deg Xi = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (ΩnR/IR/I(M),I) is a Golod pair over R where ΩnR/IR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ω¹R/IR/I(M),I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R. Finally we prove that if (M, I) is a Golod pair over R and if a ∈ I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π : R → R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps π₁ : R → R/(a) and π₂ : R/(a) → R/1 are Golod homomorphisms."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Resolutions mod I, Golod pairs"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Green, Edward L."],"dc:contributor.committeemember":["Farkas, Daniel R.","Thomson, James E.","McCoy, Robert A.","Arnold, Jimmy T."],"dc:contributor.department":["Mathematics"],"dc:creator":["Gokhale, Dhananjay R."],"dc:date.accessioned":["2014-03-14T21:19:09Z"],"dc:date.available":["2014-03-14T21:19:09Z","2005-09-20"],"dc:date.issued":["1992-04-05"],"dc:description.abstract":["Let R be a commutative ring, I be an ideal in R and let M be a R/ I -module. In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then ∐∞n=o TorR/In (M, N) is a graded module over the polynomial ring R/ I [X₁. .. Xr] with deg Xi = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (ΩnR/IR/I(M),I) is a Golod pair over R where ΩnR/IR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ω¹R/IR/I(M),I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R. Finally we prove that if (M, I) is a Golod pair over R and if a ∈ I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π : R → R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps π₁ : R → R/(a) and π₂ : R/(a) → R/1 are Golod homomorphisms."],"dc:description.degree":["Ph. D."],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-09202005-091014"],"dc:identifier.uri":["http://hdl.handle.net/10919/39431"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Resolutions mod I, Golod pairs"],"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":"2026-07-22T22:19:12Z"}