{"title":"Quillen's solution of Serre's Problem","authors":"A. Suslin","doi":"10.1017/IS012012010JKT205","DOIUrl":"https://doi.org/10.1017/IS012012010JKT205","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"549-552"},"PeriodicalIF":0.0,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS012012010JKT205","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666862","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}
{"title":"Spectrum of group cohomology and support varieties","authors":"E. Friedlander","doi":"10.1017/IS011012012JKT206","DOIUrl":"https://doi.org/10.1017/IS011012012JKT206","url":null,"abstract":"It is a great privilege to reflect on the results and influence of Daniel Quillen’s two papers in the Annals (1971) entitled “The spectrum of an equivariant cohomology ring, I, II” [26], [27]. As with other papers by Dan, these are very clearly written and reach their conclusions with elegance and efficiency. The object of study is the equivariant cohomology algebra of a compact Lie group G acting on a reasonable topological space. A case of particular interest is the action of a finite group on a point, in which case the ring in question is the cohomology algebra of the finite group. Dan writes: “It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G.” What follows is a brief introduction to Dan’s results and methods, followed by an idiosyncratic discussion of some subsequent developments. The specific goal of Dan’s first paper [26] is to give a proof of a conjecture by M. Atiyah (unpublished) and R. Swan [31] concerning the Krull dimension of the mod-p cohomology of a finite group. Those familiar with Dan’s style will not be surprised that in reaching his goal he lays out clearly and concisely the foundations for equivariant cohomology as introduced by A. Borel [7]. Although we do not address the many topological applications of equivariant cohomology or recent developments using equivariant theories in algebraic geometry, we would remiss if we did not point out that this paper establishes the definitions and techniques which a generation of mathematicians have actively pursued. The second paper [26] opens the way to many subsequent developments in the cohomology and representation theory of groups and related structures. Namely, Dan identifies up to (Zariski) homeomorphism the spectrum of the cohomology of a finite group in terms of its elementary abelian p-subgroups. More recent developments which are an outgrowth of Dan’s methods are the subject of the latter part of this survey. The lasting impact of this second paper is in large part due to its vision of investigating group theory using the commutative algebra of the cohomology of groups, leading to new roles for algebraic geometry and triangulated categories in the study of representation theory.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"148 1","pages":"507-516"},"PeriodicalIF":0.0,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86652753","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}
{"title":"Algebraic K -theory and cyclic homology","authors":"J. Loday","doi":"10.1017/IS012011006JKT200","DOIUrl":"https://doi.org/10.1017/IS012011006JKT200","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"553-557"},"PeriodicalIF":0.0,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS012011006JKT200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666836","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}
{"title":"Twisted Witt Groups of Flag Varieties","authors":"Marcus Zibrowius","doi":"10.1017/is014003028jkt260","DOIUrl":"https://doi.org/10.1017/is014003028jkt260","url":null,"abstract":"Calmes and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading. Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"14 1","pages":"139-184"},"PeriodicalIF":0.0,"publicationDate":"2013-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is014003028jkt260","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56668219","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}
{"title":"An interesting example for spectral invariants","authors":"M. Benameur, J. Heitsch, C. Wahl","doi":"10.1017/IS014002020JKT255","DOIUrl":"https://doi.org/10.1017/IS014002020JKT255","url":null,"abstract":"In [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"13 1","pages":"305-311"},"PeriodicalIF":0.0,"publicationDate":"2013-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS014002020JKT255","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56668090","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}
{"title":"Godeaux–Serre varieties and the étale index","authors":"Benjamin Antieau, B. Williams","doi":"10.1017/IS013003003JKT220","DOIUrl":"https://doi.org/10.1017/IS013003003JKT220","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"283-295"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013003003JKT220","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667912","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}
{"title":"Algebraic cycles satisfying the Maurer-Cartan equation and the unipotent fundamental group of curves","authors":"Majid Hadian","doi":"10.1017/IS013002015JKT216","DOIUrl":"https://doi.org/10.1017/IS013002015JKT216","url":null,"abstract":"We address the question of lifting the etale unipotent fundamental group of curves to the level of algebraic cycles and show that a sequence of algebraic cycles whose sum satisfies the Maurer-Cartan equation would do the job. For any elliptic curve with the origin removed and the curve , we construct such a sequence of algebraic cycles whose image under the cycle map gives rise to the etale unipotent fundamental group of the curve.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"351-392"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013002015JKT216","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667783","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}
{"title":"Injective stability for unitary K 1 , revisited","authors":"S. Sinchuk","doi":"10.1017/IS013001028JKT211","DOIUrl":"https://doi.org/10.1017/IS013001028JKT211","url":null,"abstract":"We prove the injective stability theorem for unitary K 1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger Λ-stable range condition was used.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"233-242"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013001028JKT211","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667124","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}
{"title":"Extensions panachées autoduales","authors":"D. Bertrand","doi":"10.1017/IS013001030JKT213","DOIUrl":"https://doi.org/10.1017/IS013001030JKT213","url":null,"abstract":"We study self-duality of Grothendieck's blended extensions (extensions panach'ees) in the context of a tannakian category. The set of equivalence classes of symmetric, resp. antisymmetric, blended extensions is naturally endowed with a torsor structure, which enables us to compute the unipotent radical of the associated monodromy groups in various situations","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"393-411"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013001030JKT213","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667244","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}
{"title":"The projective bundle theorem for Ij -cohomology","authors":"J. Fasel","doi":"10.1017/IS013002015JKT217","DOIUrl":"https://doi.org/10.1017/IS013002015JKT217","url":null,"abstract":"","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"11 1","pages":"413-464"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS013002015JKT217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56667852","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}