{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113270"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113270","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A motivic norm structure on equivariant algebraic K-theory","abstract":"Equivariant motivic homotopy theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\\infty$-category $\\SH^G(S)$. We use the model categorical setup to reproduce the motivic norm functors, which were originally defined in \\cite{BH}. In the second part, we define a theory of orientation in $A$-equivariant motivic homotopy theory, at least for a finite abelian group $A$. As an application, we prove the equivariant motivic analogue of the Snaith theorem $$(\\Sigma^\\infty_+ \\P(\\U_A))[\\beta^{-1}] \\simeq \\KGL_A$$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.","abstract_html":"Equivariant motivic homotopy theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\\infty$-category <span class=\"etd-inline-math\">\\SH<sup>G</sup>(S)</span>. We use the model categorical setup to reproduce the motivic norm functors, which were originally defined in \\cite{BH}. In the second part, we define a theory of orientation in $A$-equivariant motivic homotopy theory, at least for a finite abelian group $A$. As an application, we prove the equivariant motivic analogue of the Snaith theorem $<span class=\"etd-inline-math\">(\\Sigma<sup>\\</sup>infty<sub>+</sub> \\P(\\U<sub>A</sub>))[&beta;<sup>-1</sup>] \\simeq \\KGL<sub>A</sub></span>$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.","abstract_has_math":true,"creators":["Okano, Tsutomu"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Heller, Jeremiah","McCarthy, Randy","Rezk, Charles","Stojanoska, Vesna"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T22:54:09Z","date_published":"2022-01-12T22:54:09Z","updated_at":"2026-07-22T22:24:53Z","subjects":["Homotopy theory","Motivic homotopy theory","Algebraic K-theory","Algebraic geometry"],"languages":["en"],"rights":["Copyright 2021 Tsutomu Okano"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113270","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Heller, Jeremiah","McCarthy, Randy","Rezk, Charles","Stojanoska, Vesna"]},{"key":"dc:creator","label":"Author","values":["Okano, Tsutomu"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T22:54:09Z","2024-01-12T22:56:20Z","2021-07-08","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Homotopy theory","Motivic homotopy theory","Algebraic K-theory","Algebraic geometry"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021 Tsutomu Okano"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113270"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Equivariant motivic homotopy theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\\infty$-category $\\SH^G(S)$. We use the model categorical setup to reproduce the motivic norm functors, which were originally defined in \\cite{BH}. In the second part, we define a theory of orientation in $A$-equivariant motivic homotopy theory, at least for a finite abelian group $A$. As an application, we prove the equivariant motivic analogue of the Snaith theorem $$(\\Sigma^\\infty_+ \\P(\\U_A))[\\beta^{-1}] \\simeq \\KGL_A$$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2023-08-01","The student, Tsutomu Okano, accepted the attached license on 2021-07-04 at 03:42.","The student, Tsutomu Okano, submitted this Dissertation for approval on 2021-07-04 at 04:16.","This Dissertation was approved for publication on 2021-07-08 at 16:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16754 on 2022-01-12 at 13:03:53","Made available in DSpace on 2022-01-12T22:54:09Z (GMT). No. of bitstreams: 3 OKANO-DISSERTATION-2021.pdf: 759688 bytes, checksum: cf9a9bdb55ebeb7910488f5e7ff81477 (MD5) LICENSE.txt: 4210 bytes, checksum: 25aecbf21c14603a73be15d6c880607c (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: 787486f6e750d1224e004ef8dc59f2ac (MD5) Previous issue date: 2021-07-08","Embargo set by: Seth Robbins for item 121197 Lift date: 2024-01-12T22:54:14Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 121197 Lift date: 2024-01-12T22:55:09Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Embargo set by: Seth Robbins for item 121197 Lift date: 2024-01-12T22:56:20Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Author requested closed access (OA after 2yrs) in Vireo ETD system","Limited"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A motivic norm structure on equivariant algebraic K-theory"]}]}],"canonical_facts":{"dc:contributor":["Heller, Jeremiah","McCarthy, Randy","Rezk, Charles","Stojanoska, Vesna"],"dc:creator":["Okano, Tsutomu"],"dc:date":["2022-01-12T22:54:09Z","2024-01-12T22:56:20Z","2021-07-08","2021-08"],"dc:description":["Equivariant motivic homotopy theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\\infty$-category $\\SH^G(S)$. We use the model categorical setup to reproduce the motivic norm functors, which were originally defined in \\cite{BH}. In the second part, we define a theory of orientation in $A$-equivariant motivic homotopy theory, at least for a finite abelian group $A$. As an application, we prove the equivariant motivic analogue of the Snaith theorem $$(\\Sigma^\\infty_+ \\P(\\U_A))[\\beta^{-1}] \\simeq \\KGL_A$$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.","Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2023-08-01","The student, Tsutomu Okano, accepted the attached license on 2021-07-04 at 03:42.","The student, Tsutomu Okano, submitted this Dissertation for approval on 2021-07-04 at 04:16.","This Dissertation was approved for publication on 2021-07-08 at 16:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16754 on 2022-01-12 at 13:03:53","Made available in DSpace on 2022-01-12T22:54:09Z (GMT). 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