Pub Date : 2011-02-01DOI: 10.1017/IS010001012JKT098
P. Hu, I. Kríz, K. Ormsby
We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.
{"title":"Remarks on motivic homotopy theory over algebraically closed fields","authors":"P. Hu, I. Kríz, K. Ormsby","doi":"10.1017/IS010001012JKT098","DOIUrl":"https://doi.org/10.1017/IS010001012JKT098","url":null,"abstract":"We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"7 1","pages":"55-89"},"PeriodicalIF":0.0,"publicationDate":"2011-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010001012JKT098","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-02-01DOI: 10.1017/IS010001012JKT100
R. Sperber
{"title":"Comparing Assembly Maps in Algebraic K -Theory","authors":"R. Sperber","doi":"10.1017/IS010001012JKT100","DOIUrl":"https://doi.org/10.1017/IS010001012JKT100","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"7 1","pages":"145-168"},"PeriodicalIF":0.0,"publicationDate":"2011-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS010001012JKT100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-02-01DOI: 10.1017/IS009010019JKT081
D. Simson, D. Simson
{"title":"The Euler characteristic and Euler defect for comodules over Euler coalgebras","authors":"D. Simson, D. Simson","doi":"10.1017/IS009010019JKT081","DOIUrl":"https://doi.org/10.1017/IS009010019JKT081","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"7 1","pages":"91-113"},"PeriodicalIF":0.0,"publicationDate":"2011-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009010019JKT081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-01-10DOI: 10.1017/is011010005jkt168
Otgonbayar Uuye
We extend McClure's results regarding restriction maps in equivariant K-theory to bivariant K-theory: Let G be a compact Lie group and A and B be G-C*-algebras. Suppose that KKHn (A, B) is a finitely generated R(G)-module for every H ≤ G closed and n ∈ ℤ. Then, if KKF*(A, B) = 0 for all F ≤ G finite cyclic, then KKG*(A, B) = 0.
我们将等变k理论中关于限制映射的McClure的结果推广到二变k理论:设G是紧李群，a和B是G- c *代数。设KKHn (A, B)是一个有限生成的R(G)-模，对于每一个H≤G闭且n∈n。则对于所有F≤G有限循环，若KKF*(A, B) = 0，则KKG*(A, B) = 0。
{"title":"Restriction maps in equivariant KK-theory","authors":"Otgonbayar Uuye","doi":"10.1017/is011010005jkt168","DOIUrl":"https://doi.org/10.1017/is011010005jkt168","url":null,"abstract":"We extend McClure's results regarding restriction maps in equivariant K-theory to bivariant K-theory: \u0000Let G be a compact Lie group and A and B be G-C*-algebras. Suppose that KKHn (A, B) is a finitely generated R(G)-module for every H ≤ G closed and n ∈ ℤ. Then, if KKF*(A, B) = 0 for all F ≤ G finite cyclic, then KKG*(A, B) = 0.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"45-55"},"PeriodicalIF":0.0,"publicationDate":"2011-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is011010005jkt168","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-12-18DOI: 10.1017/is011010026jkt167
G. Modoi, J. Šťovíček
We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.
{"title":"Brown representability often fails for homotopy categories of complexes","authors":"G. Modoi, J. Šťovíček","doi":"10.1017/is011010026jkt167","DOIUrl":"https://doi.org/10.1017/is011010026jkt167","url":null,"abstract":"We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"151-160"},"PeriodicalIF":0.0,"publicationDate":"2010-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is011010026jkt167","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-12-01DOI: 10.1017/IS008008012JKT084
N. Yagita
{"title":"Motivic cohomology of quadrics and the coniveau spectral sequence","authors":"N. Yagita","doi":"10.1017/IS008008012JKT084","DOIUrl":"https://doi.org/10.1017/IS008008012JKT084","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"6 1","pages":"547-589"},"PeriodicalIF":0.0,"publicationDate":"2010-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS008008012JKT084","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-12-01DOI: 10.1017/IS009010008JKT094
Denis-Charles Cisinski
The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy equivalence between the corresponding K -theory spectra. This generalizes a well known result of Thomason and Trobaugh. The ingredients, for this proof, are a generalization of the Waldhausen approximation theorem, and a simple combinatorial caracterization of derived equivalences. We also study simplicial localization of Waldhausen categories. We prove that a (homotopy) right exact functor induces an equivalence of homotopy categories if and only if it induces an equivalence of simplicial localizations. This allows to make the link with the K -theory of simplicial categories introduced by Toen and Vezzosi.
{"title":"Invariance de la K-théorie par équivalences dérivées","authors":"Denis-Charles Cisinski","doi":"10.1017/IS009010008JKT094","DOIUrl":"https://doi.org/10.1017/IS009010008JKT094","url":null,"abstract":"The aim of these notes is to prove that any right exact functor between reasonable Waldhausen categories, that induces an equivalence at the level of homotopy categories, gives rise to a homotopy equivalence between the corresponding K -theory spectra. This generalizes a well known result of Thomason and Trobaugh. The ingredients, for this proof, are a generalization of the Waldhausen approximation theorem, and a simple combinatorial caracterization of derived equivalences. We also study simplicial localization of Waldhausen categories. We prove that a (homotopy) right exact functor induces an equivalence of homotopy categories if and only if it induces an equivalence of simplicial localizations. This allows to make the link with the K -theory of simplicial categories introduced by Toen and Vezzosi.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"188 1","pages":"505-546"},"PeriodicalIF":0.0,"publicationDate":"2010-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009010008JKT094","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-11-18DOI: 10.1017/is011004017jkt156
T. M. Carlsen, S. Eilers, M. Tomforde
Let C � (E) be the graph C � -algebra associated to a graph E and let J be a gauge-invariant ideal in C � (E). We compute the cyclic six-term exact sequence in K-theory associated to the extension 0 ! J ! C � (E) ! C � (E)/J ! 0 in terms of the adjacency matrix associated to E. The ordered six- term exact sequence is a complete stable isomorphism invariant for se- veral classes of graph C � -algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences comprise complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.
{"title":"INDEX MAPS IN THE K-THEORY OF GRAPH ALGEBRAS","authors":"T. M. Carlsen, S. Eilers, M. Tomforde","doi":"10.1017/is011004017jkt156","DOIUrl":"https://doi.org/10.1017/is011004017jkt156","url":null,"abstract":"Let C � (E) be the graph C � -algebra associated to a graph E and let J be a gauge-invariant ideal in C � (E). We compute the cyclic six-term exact sequence in K-theory associated to the extension 0 ! J ! C � (E) ! C � (E)/J ! 0 in terms of the adjacency matrix associated to E. The ordered six- term exact sequence is a complete stable isomorphism invariant for se- veral classes of graph C � -algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences comprise complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"385-406"},"PeriodicalIF":0.0,"publicationDate":"2010-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is011004017jkt156","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56665795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-10-01DOI: 10.1017/IS009012013JKT079
A. Dooms, E. Jespers, A. Konovalov
{"title":"From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups","authors":"A. Dooms, E. Jespers, A. Konovalov","doi":"10.1017/IS009012013JKT079","DOIUrl":"https://doi.org/10.1017/IS009012013JKT079","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"54 1","pages":"263-283"},"PeriodicalIF":0.0,"publicationDate":"2010-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009012013JKT079","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56663129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2010-10-01DOI: 10.1017/IS009010003JKT092
A. Carey, J. Phillips, A. Rennie
This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K 1 -group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.
{"title":"Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra O n","authors":"A. Carey, J. Phillips, A. Rennie","doi":"10.1017/IS009010003JKT092","DOIUrl":"https://doi.org/10.1017/IS009010003JKT092","url":null,"abstract":"This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K 1 -group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"6 1","pages":"339-380"},"PeriodicalIF":0.0,"publicationDate":"2010-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009010003JKT092","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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