{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21180"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21180","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Automorphisms and symbols in K(,2)","abstract":"\"Much of the work in algebraic K-theory today is devoted to the search for \"\"motivic cohomology.\"\" This hoped-for cohomology theory of algebraic geometry should be analogous to the known singular homology groups of topology. In this work we consider Goodwillie and Lichtenbaum's candidate for motivic cohomology. For R a regular ring, they define a \"\"weight\"\" filtration on the K-theory space K(R),$$K(R) = W\\sp0 \\gets W\\sp1 \\gets W\\sp2 \\gets \\cdots,$$and then define the cohomology groups to be$$H\\sp{m}(X,\\doubz (t)) = \\pi\\sb{2t-m}(W\\sp{t}/W\\sp{t+1}).$$If what they propose is correct, then one would expect $\\pi\\sb{t}(W\\sp{t}/W\\sp{t+l})$ to be the weight t part of K(R). Here we present Goodwillie's proof that this is indeed the case when t = 2 and R is an algebraically closed field. We then continue his work by showing that $\\pi\\sb2(W\\sp2/W\\sp3)$ $\\cong$ K$\\sb2(R)$ for other fields, namely, we handle the cases, R = $\\IR$, and R = $\\doubc((T)).$ Our arguments also work when R is a finite field.\"","abstract_html":"&quot;Much of the work in algebraic K-theory today is devoted to the search for &quot;&quot;motivic cohomology.&quot;&quot; This hoped-for cohomology theory of algebraic geometry should be analogous to the known singular homology groups of topology. In this work we consider Goodwillie and Lichtenbaum&#x27;s candidate for motivic cohomology. For R a regular ring, they define a &quot;&quot;weight&quot;&quot; filtration on the K-theory space K(R),$$K(R) = W\\sp0 \\gets W\\sp1 \\gets W\\sp2 \\gets \\cdots,$$and then define the cohomology groups to be$<span class=\"etd-inline-math\">H\\sp{m}(X,\\doubz (t)) = &pi;\\sb{2t-m}(W\\sp{t}/W\\sp{t+1}).</span>$If what they propose is correct, then one would expect $\\pi\\sb{t}(W\\sp{t}/W\\sp{t+l})$ to be the weight t part of K(R). Here we present Goodwillie&#x27;s proof that this is indeed the case when t = 2 and R is an algebraically closed field. We then continue his work by showing that $\\pi\\sb2(W\\sp2/W\\sp3)$ $\\cong$ K$\\sb2(R)$ for other fields, namely, we handle the cases, R = $\\IR$, and R = $\\doubc((T)).$ Our arguments also work when R is a finite field.&quot;","abstract_has_math":true,"creators":["Holdener, Judy Ann"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Evans, Graham"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:00:47Z","date_published":"2011-05-07T13:00:47Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Holdener, Judy Ann"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512397","(UMI)AAI9512397"],"render_values":[{"text":"AAI9512397","href":null,"code":true},{"text":"(UMI)AAI9512397","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21180","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Evans, Graham"]},{"key":"dc:creator","label":"Author","values":["Holdener, Judy Ann"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:00:47Z","10000-01-01","1994"]},{"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"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Holdener, Judy Ann"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9512397","(UMI)AAI9512397","http://hdl.handle.net/2142/21180"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Much of the work in algebraic K-theory today is devoted to the search for \"\"motivic cohomology.\"\" This hoped-for cohomology theory of algebraic geometry should be analogous to the known singular homology groups of topology. In this work we consider Goodwillie and Lichtenbaum's candidate for motivic cohomology. For R a regular ring, they define a \"\"weight\"\" filtration on the K-theory space K(R),$$K(R) = W\\sp0 \\gets W\\sp1 \\gets W\\sp2 \\gets \\cdots,$$and then define the cohomology groups to be$$H\\sp{m}(X,\\doubz (t)) = \\pi\\sb{2t-m}(W\\sp{t}/W\\sp{t+1}).$$If what they propose is correct, then one would expect $\\pi\\sb{t}(W\\sp{t}/W\\sp{t+l})$ to be the weight t part of K(R). Here we present Goodwillie's proof that this is indeed the case when t = 2 and R is an algebraically closed field. We then continue his work by showing that $\\pi\\sb2(W\\sp2/W\\sp3)$ $\\cong$ K$\\sb2(R)$ for other fields, namely, we handle the cases, R = $\\IR$, and R = $\\doubc((T)).$ Our arguments also work when R is a finite field.\"","Made available in DSpace on 2011-05-07T13:00:47Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512397.pdf: 1798472 bytes, checksum: c2365638dec74d9180f54e1c6ff1a18e (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:01Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:15-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Automorphisms and symbols in K(,2)"]}]}],"canonical_facts":{"dc:contributor":["Evans, Graham"],"dc:creator":["Holdener, Judy Ann"],"dc:date":["2011-05-07T13:00:47Z","10000-01-01","1994"],"dc:description":["\"Much of the work in algebraic K-theory today is devoted to the search for \"\"motivic cohomology.\"\" This hoped-for cohomology theory of algebraic geometry should be analogous to the known singular homology groups of topology. In this work we consider Goodwillie and Lichtenbaum's candidate for motivic cohomology. For R a regular ring, they define a \"\"weight\"\" filtration on the K-theory space K(R),$$K(R) = W\\sp0 \\gets W\\sp1 \\gets W\\sp2 \\gets \\cdots,$$and then define the cohomology groups to be$$H\\sp{m}(X,\\doubz (t)) = \\pi\\sb{2t-m}(W\\sp{t}/W\\sp{t+1}).$$If what they propose is correct, then one would expect $\\pi\\sb{t}(W\\sp{t}/W\\sp{t+l})$ to be the weight t part of K(R). Here we present Goodwillie's proof that this is indeed the case when t = 2 and R is an algebraically closed field. We then continue his work by showing that $\\pi\\sb2(W\\sp2/W\\sp3)$ $\\cong$ K$\\sb2(R)$ for other fields, namely, we handle the cases, R = $\\IR$, and R = $\\doubc((T)).$ Our arguments also work when R is a finite field.\"","Made available in DSpace on 2011-05-07T13:00:47Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9512397.pdf: 1798472 bytes, checksum: c2365638dec74d9180f54e1c6ff1a18e (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:49:01Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:22:15-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9512397","(UMI)AAI9512397","http://hdl.handle.net/2142/21180"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Holdener, Judy Ann"],"dc:subject":["Mathematics"],"dc:title":["Automorphisms and symbols in K(,2)"],"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":"2026-07-22T22:25:17Z"}