{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86978"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86978","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Adams Operations and the Dennis Trace Map","abstract":"For a commutative algebra A, the algebraic K-theory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic K-theory, one would like to use the Dennis trace map to study the algebraic K-theory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic K-theory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on K-theory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the S-construction model for K-theory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the S-construction for K-theory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology.","abstract_html":"For a commutative algebra A, the algebraic K-theory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic K-theory, one would like to use the Dennis trace map to study the algebraic K-theory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic K-theory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on K-theory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the S-construction model for K-theory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the S-construction for K-theory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology.","abstract_has_math":false,"creators":["Kantorovitz, Miriam Ruth"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["McCarthy, Randy"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:26Z","date_published":"2015-09-28T15:20:26Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9944906"],"render_values":[{"text":"(MiAaPQ)AAI9944906","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86978","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McCarthy, Randy"]},{"key":"dc:creator","label":"Author","values":["Kantorovitz, Miriam Ruth"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:26Z","10000-01-01","1999"]},{"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":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86978","(MiAaPQ)AAI9944906"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["For a commutative algebra A, the algebraic K-theory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic K-theory, one would like to use the Dennis trace map to study the algebraic K-theory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic K-theory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on K-theory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the S-construction model for K-theory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the S-construction for K-theory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology.","Made available in DSpace on 2015-09-28T15:20:26Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944906.pdf: 1590400 bytes, checksum: b51d65c270647ce17c5664f1143dbfd1 (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88259 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","47 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."]},{"key":"dc:title","label":"Title","values":["Adams Operations and the Dennis Trace Map"]}]}],"canonical_facts":{"dc:contributor":["McCarthy, Randy"],"dc:creator":["Kantorovitz, Miriam Ruth"],"dc:date":["2015-09-28T15:20:26Z","10000-01-01","1999"],"dc:description":["For a commutative algebra A, the algebraic K-theory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic K-theory, one would like to use the Dennis trace map to study the algebraic K-theory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic K-theory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on K-theory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the S-construction model for K-theory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the S-construction for K-theory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology.","Made available in DSpace on 2015-09-28T15:20:26Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9944906.pdf: 1590400 bytes, checksum: b51d65c270647ce17c5664f1143dbfd1 (MD5) Previous issue date: 1999","Embargo set by: Seth Robbins for item 88259 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","47 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999."],"dc:identifier":["http://hdl.handle.net/2142/86978","(MiAaPQ)AAI9944906"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Adams Operations and the Dennis Trace Map"],"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":"2026-07-22T22:26:28Z"}