{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/29259"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/29259","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"Torsors over Simplicial Schemes","abstract":"Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, as well as a characterization of the second sheaf cohomology group of BG with coefficients in a sheaf of abelian groups A, in terms of central extensions of G by A. It is well-known that, if k is a perfect field, the motivic cohomology of the classifying space BGL_n of the general linear group is a polynomial algebra over the motivic cohomology of k; we give a proof that takes advantage of this theory of torsors over simplicial schemes. Finally, using the work of Vistoli, we prove that, working over the complex numbers, the map from the Chow groups of the etale classifying space of the projective linear group PGL_p to the Chow groups of the Nisnevich classifying space of PGL_p is injective, when p is an odd prime.","abstract_html":"Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, as well as a characterization of the second sheaf cohomology group of BG with coefficients in a sheaf of abelian groups A, in terms of central extensions of G by A. It is well-known that, if k is a perfect field, the motivic cohomology of the classifying space BGL_n of the general linear group is a polynomial algebra over the motivic cohomology of k; we give a proof that takes advantage of this theory of torsors over simplicial schemes. Finally, using the work of Vistoli, we prove that, working over the complex numbers, the map from the Chow groups of the etale classifying space of the projective linear group PGL_p to the Chow groups of the Nisnevich classifying space of PGL_p is injective, when p is an odd prime.","abstract_has_math":false,"creators":["Rolle, Alexander S"],"institution":"The University of Western Ontario","degree_name":"Ph D","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Jardine, John F."],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-08-08","date_published":"2019-08-08","updated_at":"2026-07-27T21:56:05Z","subjects":["Torsors","Simplicial Schemes","Motivic Cohomology"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/29259","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Jardine, John F."]},{"key":"dc:creator","label":"Author","values":["Rolle, Alexander S"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T16:18:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T16:18:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-08-08"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph D"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Torsors","Simplicial Schemes","Motivic Cohomology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/29259"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, as well as a characterization of the second sheaf cohomology group of BG with coefficients in a sheaf of abelian groups A, in terms of central extensions of G by A. It is well-known that, if k is a perfect field, the motivic cohomology of the classifying space BGL_n of the general linear group is a polynomial algebra over the motivic cohomology of k; we give a proof that takes advantage of this theory of torsors over simplicial schemes. Finally, using the work of Vistoli, we prove that, working over the complex numbers, the map from the Chow groups of the etale classifying space of the projective linear group PGL_p to the Chow groups of the Nisnevich classifying space of PGL_p is injective, when p is an odd prime."]},{"key":"dc:title","label":"Title","values":["Torsors over Simplicial Schemes"]}]}],"canonical_facts":{"dc:contributor.advisor":["Jardine, John F."],"dc:creator":["Rolle, Alexander S"],"dc:date.accessioned":["2025-07-10T16:18:56Z"],"dc:date.available":["2025-07-10T16:18:56Z"],"dc:date.issued":["2019-08-08"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.abstract":["Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, as well as a characterization of the second sheaf cohomology group of BG with coefficients in a sheaf of abelian groups A, in terms of central extensions of G by A. It is well-known that, if k is a perfect field, the motivic cohomology of the classifying space BGL_n of the general linear group is a polynomial algebra over the motivic cohomology of k; we give a proof that takes advantage of this theory of torsors over simplicial schemes. Finally, using the work of Vistoli, we prove that, working over the complex numbers, the map from the Chow groups of the etale classifying space of the projective linear group PGL_p to the Chow groups of the Nisnevich classifying space of PGL_p is injective, when p is an odd prime."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/29259"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["Torsors","Simplicial Schemes","Motivic Cohomology"],"dc:title":["Torsors over Simplicial Schemes"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Ph D"]},"updated_at":"2026-07-27T21:56:05Z"}