{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72542"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72542","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Weak Purity for Gorenstein Rings","abstract":"In the language of local algebra, the classical purity of branch locus theorem states that a module finite ring extension of local normal domains $B\\to A$ which is unramified in codimension one, with B a regular local ring, is unramified (and in this setting, etale). When the ring B is merely Gorenstein, there are numerous examples which show that A need not be Cohen-Macaulay, hence that the extension need not be unramified. In a related context, the main portion of the thesis addresses the purity of the extension $B\\to A.$ The setting is as follows: $B\\to A$ is a module finite extension of normal rings, unramified in codimension one with B an excellent local equicharacteristic Gorenstein domain of dimension at least five (under additional conditions, the mixed characteristic case is considered). In this context, a weak purity holds: if B is "regular" enough (that is, satisfies $(R\\sb{k})),$ then A inherits a certain amount of depth (that is, satisfies $(S\\sb{k-1}))$ where $k\\ge 4.$ This purity is weak in that A acquires "good" properties from B, yet the extension itself need not be "good" (that is, unramified), as is illustrated by an example. Moreover, the method of proof is used to recover Grothendieck's purity theorem for complete intersections in the special case of a hypersurface ring B. Applications are considered: in extensions $B\\to A$ similar to those above which are normal (with Galois group G), a relationship between codimension two primes of A which are fixed under the action of G and small MCM A-modules (via "Bourbaki"-exact sequences) is examined; depth properties of divisorial B-ideals of finite order in Cl(B) are investigated; and related ideas are studied.","abstract_html":"In the language of local algebra, the classical purity of branch locus theorem states that a module finite ring extension of local normal domains $B\\to A$ which is unramified in codimension one, with B a regular local ring, is unramified (and in this setting, etale). When the ring B is merely Gorenstein, there are numerous examples which show that A need not be Cohen-Macaulay, hence that the extension need not be unramified. In a related context, the main portion of the thesis addresses the purity of the extension $B\\to A.$ The setting is as follows: $B\\to A$ is a module finite extension of normal rings, unramified in codimension one with B an excellent local equicharacteristic Gorenstein domain of dimension at least five (under additional conditions, the mixed characteristic case is considered). In this context, a weak purity holds: if B is "regular" enough (that is, satisfies $(R\\sb{k})),$ then A inherits a certain amount of depth (that is, satisfies $(S\\sb{k-1}))$ where $k\\ge 4.$ This purity is weak in that A acquires "good" properties from B, yet the extension itself need not be "good" (that is, unramified), as is illustrated by an example. Moreover, the method of proof is used to recover Grothendieck's purity theorem for complete intersections in the special case of a hypersurface ring B. Applications are considered: in extensions $B\\to A$ similar to those above which are normal (with Galois group G), a relationship between codimension two primes of A which are fixed under the action of G and small MCM A-modules (via "Bourbaki"-exact sequences) is examined; depth properties of divisorial B-ideals of finite order in Cl(B) are investigated; and related ideas are studied.","abstract_has_math":true,"creators":["Borek, Adam Richard"],"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":2014,"date_issued":"2014-12-17T23:17:47Z","date_published":"2014-12-17T23:17:47Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9411569"],"render_values":[{"text":"(UMI)AAI9411569","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72542","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":["Borek, Adam Richard"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:47Z","10000-01-01","1993"]},{"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/2142/72542","(UMI)AAI9411569"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In the language of local algebra, the classical purity of branch locus theorem states that a module finite ring extension of local normal domains $B\\to A$ which is unramified in codimension one, with B a regular local ring, is unramified (and in this setting, etale). When the ring B is merely Gorenstein, there are numerous examples which show that A need not be Cohen-Macaulay, hence that the extension need not be unramified. In a related context, the main portion of the thesis addresses the purity of the extension $B\\to A.$ The setting is as follows: $B\\to A$ is a module finite extension of normal rings, unramified in codimension one with B an excellent local equicharacteristic Gorenstein domain of dimension at least five (under additional conditions, the mixed characteristic case is considered). In this context, a weak purity holds: if B is "regular" enough (that is, satisfies $(R\\sb{k})),$ then A inherits a certain amount of depth (that is, satisfies $(S\\sb{k-1}))$ where $k\\ge 4.$ This purity is weak in that A acquires "good" properties from B, yet the extension itself need not be "good" (that is, unramified), as is illustrated by an example. Moreover, the method of proof is used to recover Grothendieck's purity theorem for complete intersections in the special case of a hypersurface ring B. Applications are considered: in extensions $B\\to A$ similar to those above which are normal (with Galois group G), a relationship between codimension two primes of A which are fixed under the action of G and small MCM A-modules (via "Bourbaki"-exact sequences) is examined; depth properties of divisorial B-ideals of finite order in Cl(B) are investigated; and related ideas are studied.","Made available in DSpace on 2014-12-17T23:17:47Z (GMT). No. of bitstreams: 1 9411569.pdf: 2108069 bytes, checksum: f367f2aa30813ffb3c11f64b6fed2540 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72710 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","66 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."]},{"key":"dc:title","label":"Title","values":["Weak Purity for Gorenstein Rings"]}]}],"canonical_facts":{"dc:contributor":["Griffith, Phillip A."],"dc:creator":["Borek, Adam Richard"],"dc:date":["2014-12-17T23:17:47Z","10000-01-01","1993"],"dc:description":["In the language of local algebra, the classical purity of branch locus theorem states that a module finite ring extension of local normal domains $B\\to A$ which is unramified in codimension one, with B a regular local ring, is unramified (and in this setting, etale). When the ring B is merely Gorenstein, there are numerous examples which show that A need not be Cohen-Macaulay, hence that the extension need not be unramified. In a related context, the main portion of the thesis addresses the purity of the extension $B\\to A.$ The setting is as follows: $B\\to A$ is a module finite extension of normal rings, unramified in codimension one with B an excellent local equicharacteristic Gorenstein domain of dimension at least five (under additional conditions, the mixed characteristic case is considered). In this context, a weak purity holds: if B is "regular" enough (that is, satisfies $(R\\sb{k})),$ then A inherits a certain amount of depth (that is, satisfies $(S\\sb{k-1}))$ where $k\\ge 4.$ This purity is weak in that A acquires "good" properties from B, yet the extension itself need not be "good" (that is, unramified), as is illustrated by an example. Moreover, the method of proof is used to recover Grothendieck's purity theorem for complete intersections in the special case of a hypersurface ring B. Applications are considered: in extensions $B\\to A$ similar to those above which are normal (with Galois group G), a relationship between codimension two primes of A which are fixed under the action of G and small MCM A-modules (via "Bourbaki"-exact sequences) is examined; depth properties of divisorial B-ideals of finite order in Cl(B) are investigated; and related ideas are studied.","Made available in DSpace on 2014-12-17T23:17:47Z (GMT). No. of bitstreams: 1 9411569.pdf: 2108069 bytes, checksum: f367f2aa30813ffb3c11f64b6fed2540 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72710 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","66 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."],"dc:identifier":["http://hdl.handle.net/2142/72542","(UMI)AAI9411569"],"dc:subject":["Mathematics"],"dc:title":["Weak Purity for Gorenstein Rings"],"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":"2026-07-22T22:26:07Z"}