Pub Date : 2013-06-01DOI: 10.1017/IS012012010JKT205
A. Suslin
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Pub Date : 2013-06-01DOI: 10.1017/IS011012012JKT206
E. Friedlander
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 The specific goal of Dan’s first paper [26] is to give a proof of a conjecture by M. Atiyah (unpublished) and R. Swan [31] concerning the Krull dimension of the mod-p cohomology of a finite group.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.
很荣幸能够回顾Daniel Quillen在Annals(1971)上发表的题为“the spectrum of an equivariant上同环，I, II”[26]，[27]的两篇论文的成果和影响。与Dan的其他论文一样，这些论文写得非常清楚，并以优雅和高效的方式得出结论。研究作用于合理拓扑空间上的紧李群G的等变上同调代数。一个特别有趣的例子是有限群对一点的作用，在这种情况下，所讨论的环是有限群的上同调代数。Dan写道:“这一系列论文的目的是通过交换代数将附加在这样一个环上的不变量与g的初等阿贝尔p-子群族联系起来。”下面是对Dan的结果和方法的简要介绍，然后是对一些后续发展的特殊讨论。Dan的第一篇论文[26]的具体目标是证明M. Atiyah(未发表)和R. Swan[31]关于有限群的模p上同调的Krull维的猜想。熟悉Dan风格的人不会感到惊讶，在达到他的目标时，他清晰而简洁地阐述了A. Borel[7]引入的等变上同论的基础。虽然我们没有讨论等变上同调的许多拓扑应用或代数几何中使用等变理论的最新发展，但如果我们没有指出本文建立了一代数学家积极追求的定义和技术，我们将会有所遗漏。第二篇论文为群和相关结构的上同调和表示理论的许多后续发展开辟了道路。即，Dan用有限群的初等阿贝尔p-子群来标识其上同态的谱直至(Zariski)同胚。最近的发展是丹的方法的产物，是本调查后一部分的主题。这第二篇论文的持久影响在很大程度上是由于它使用群的上同调的交换代数来研究群论的愿景，导致代数几何和三角范畴在表征理论研究中的新角色。
{"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}
Pub Date : 2013-06-01DOI: 10.1017/IS012011006JKT200
J. Loday
{"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}
Pub Date : 2013-05-28DOI: 10.1017/is014003028jkt260
Marcus Zibrowius
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.
{"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}
Pub Date : 2013-04-26DOI: 10.1017/IS014002020JKT255
M. Benameur, J. Heitsch, C. Wahl
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.
{"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}
Pub Date : 2013-04-01DOI: 10.1017/IS013003003JKT220
Benjamin Antieau, B. Williams
{"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}
Pub Date : 2013-04-01DOI: 10.1017/IS013002015JKT216
Majid Hadian
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.
{"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}
Pub Date : 2013-04-01DOI: 10.1017/IS013001028JKT211
S. Sinchuk
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.
在一般的稳定范围条件下，证明了一元k1的内射稳定性定理。这改进了a . Bak, V. Petrov和G. Tang的稳定性定理，其中使用了更强的Λ-stable极差条件。
{"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}
Pub Date : 2013-04-01DOI: 10.1017/IS013001030JKT213
D. Bertrand
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
{"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}
Pub Date : 2013-04-01DOI: 10.1017/IS013002015JKT217
J. Fasel
{"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}
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