Pub Date : 2010-08-01DOI: 10.1017/IS009012003JKT078
M. Gomez-Aparicio
Let G be a locally compact group and ρ a non-unitary finite dimensional representation of G. We consider tensor products of ρ by some unitary representations of G in order to define two Banach algebras analogous to the group C∗-algebras, C∗(G) and C∗ r (G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture. Table des matieres Introduction 2 1. Algebres de groupe tordues 6 1.1. Definitions et proprietes principales 6 1.2. Fonctorialite 10 2. Morphisme de Baum-Connes tordu 11 2.1. Fleche de descente tordue 11 2.2. Fonctorialite 20 2.3. Descente et action de KK sur la K-theorie. 25 2.4. Construction du morphisme tordu 27 2.5. Compatibilite avec la somme directe de representations 28 3. Groupes admettant un element γ de Kasparov 33 3.1. Coefficients dans une algebre propre 33 3.2. Element γ de Kasparov 37 References 39 2000 Mathematics Subject Classification. 22D12, 22D15, 46L80, 19K35.
设G是一个局部紧群，ρ是G的非酉有限维表示。我们用G的一些酉表示来考虑ρ的张量积，以便定义两个类似于群C∗-代数，C∗(G)和C∗r (G)的Banach代数。我们计算了一类满足baum - cones猜想的群的这类代数的k理论。表3材料简介2群代数定理6 1.1。定义和所有权原则6功能型10 2。Morphisme de Baum-Connes tordu 11 2.1。在11月22日下降。功能化20 2.3。KK的下降和作用取决于k理论。25 2.4。构词法27 . 2.5。相容的平均数有一些直接的表示。群组辅助元素γ de Kasparov 33 3.1。系数是一个代数表达式33 3.2。元素γ de Kasparov 37参考文献39 2000数学学科分类。22D12, 22D15, 46L80, 19K35。
{"title":"Morphisme de Baum-Connes tordu par une représentation non unitaire","authors":"M. Gomez-Aparicio","doi":"10.1017/IS009012003JKT078","DOIUrl":"https://doi.org/10.1017/IS009012003JKT078","url":null,"abstract":"Let G be a locally compact group and ρ a non-unitary finite dimensional representation of G. We consider tensor products of ρ by some unitary representations of G in order to define two Banach algebras analogous to the group C∗-algebras, C∗(G) and C∗ r (G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture. Table des matieres Introduction 2 1. Algebres de groupe tordues 6 1.1. Definitions et proprietes principales 6 1.2. Fonctorialite 10 2. Morphisme de Baum-Connes tordu 11 2.1. Fleche de descente tordue 11 2.2. Fonctorialite 20 2.3. Descente et action de KK sur la K-theorie. 25 2.4. Construction du morphisme tordu 27 2.5. Compatibilite avec la somme directe de representations 28 3. Groupes admettant un element γ de Kasparov 33 3.1. Coefficients dans une algebre propre 33 3.2. Element γ de Kasparov 37 References 39 2000 Mathematics Subject Classification. 22D12, 22D15, 46L80, 19K35.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"6 1","pages":"23-68"},"PeriodicalIF":0.0,"publicationDate":"2010-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009012003JKT078","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56663074","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 : 2010-08-01DOI: 10.1017/IS008008012JKT090
M. Clancy, G. Ellis
We begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.
{"title":"Homology of some Artin and twisted Artin Groups","authors":"M. Clancy, G. Ellis","doi":"10.1017/IS008008012JKT090","DOIUrl":"https://doi.org/10.1017/IS008008012JKT090","url":null,"abstract":"We begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"6 1","pages":"171-196"},"PeriodicalIF":0.0,"publicationDate":"2010-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS008008012JKT090","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662156","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 : 2010-08-01DOI: 10.1017/IS009010011JKT095
P. Webb
We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.
{"title":"Stratifications and Mackey Functors II: Globally Defined Mackey Functors","authors":"P. Webb","doi":"10.1017/IS009010011JKT095","DOIUrl":"https://doi.org/10.1017/IS009010011JKT095","url":null,"abstract":"We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"6 1","pages":"99-170"},"PeriodicalIF":0.0,"publicationDate":"2010-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/IS009010011JKT095","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56662875","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 : 2010-06-08DOI: 10.1017/is011006030jkt162
Olivier Haution
We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p . These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.
{"title":"Reduced Steenrod operations and resolution of singularities","authors":"Olivier Haution","doi":"10.1017/is011006030jkt162","DOIUrl":"https://doi.org/10.1017/is011006030jkt162","url":null,"abstract":"We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p . These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"9 1","pages":"269-290"},"PeriodicalIF":0.0,"publicationDate":"2010-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/is011006030jkt162","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56666365","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 : 2010-06-01DOI: 10.1017/is010006004jkt124
A. Bak, Jonathan Rosenberg, C. Weibel
It is a great pleasure for the Editors of the Journal of K-Theory to congratulate their fellow Editor, colleague and friend Andrei Suslin on his 60th birthday and to wish him many more birthdays to come. For this occasion, the Journal of K-Theory is following a tradition begun by the earlier journal K-Theory to honor A. Grothendieck and D. Quillen on their 60th birthdays and is bringing out 2 special issues in honor of Andrei. This is the first issue. The second issue will appear later this year. We are also pleased to announce that the journal Documenta Mathematica will also honor Andrei this year with a volume of its journal. Andrei’s achievements and influence on the development of subjects served by our journal are enormous. We mention just a few of the highlights. His early work in the period 1974–1981 concerned projective, symplectic, and orthogonal modules and their automorphism groups. His solution in 1976 of the Serre conjecture on the freeness of projective modules over a polynomial ring over a field, established independently by Daniel Quillen in the same year, earned him the Komsomol Prize, the most prestigious honor for young scientists in the former Soviet Union. His paper also contained the analogous result for symplectic modules. One does not expect the same result for quadratic modules over a polynomial ring over a field (of characteristic not equal 2), but rather that such modules are extended. This result was proved in a joint paper with his student V.I. Kopeiko in 1977, under the condition that the Witt index of the module is at least 2. All of the above results are closely tied to the cancellation problem for modules over commutative rings and this was the topic of Andrei’s talk at ICM 1978 in Helsinki. It would be the first of 3 ICM talks he was invited to give, the second of which would be a plenary talk. In the years immediately after the Helsinki meeting, his work included the normality of the elementary subgroup of a general linear group of rank at least 3 over a module finite ring (one which is finitely generated as a module over its center), the centrality of any Steinberg group extension of rank at least 4 over a module finite ring (joint with his student M.S. Tulenbaev), and his stability theorem for higher K-groups over rings of finite stable rank. The first and last paper of the current issue use ideas, techniques, and results from the early period above. At the beginning of the 1980s, Andrei turned his attention to the K-theory of fields and division rings. By a classical result of Kummer, the norm residue homomorphism of degree 1 was known to be an isomorphism. In 1982, Andrei showed together with A. Merkurjev that the norm residue homomorphism of degree
{"title":"Foreword to the Special Issues in honor of Andrei Suslin on his 60th birthday","authors":"A. Bak, Jonathan Rosenberg, C. Weibel","doi":"10.1017/is010006004jkt124","DOIUrl":"https://doi.org/10.1017/is010006004jkt124","url":null,"abstract":"It is a great pleasure for the Editors of the Journal of K-Theory to congratulate their fellow Editor, colleague and friend Andrei Suslin on his 60th birthday and to wish him many more birthdays to come. For this occasion, the Journal of K-Theory is following a tradition begun by the earlier journal K-Theory to honor A. Grothendieck and D. Quillen on their 60th birthdays and is bringing out 2 special issues in honor of Andrei. This is the first issue. The second issue will appear later this year. We are also pleased to announce that the journal Documenta Mathematica will also honor Andrei this year with a volume of its journal. Andrei’s achievements and influence on the development of subjects served by our journal are enormous. We mention just a few of the highlights. His early work in the period 1974–1981 concerned projective, symplectic, and orthogonal modules and their automorphism groups. His solution in 1976 of the Serre conjecture on the freeness of projective modules over a polynomial ring over a field, established independently by Daniel Quillen in the same year, earned him the Komsomol Prize, the most prestigious honor for young scientists in the former Soviet Union. His paper also contained the analogous result for symplectic modules. One does not expect the same result for quadratic modules over a polynomial ring over a field (of characteristic not equal 2), but rather that such modules are extended. This result was proved in a joint paper with his student V.I. Kopeiko in 1977, under the condition that the Witt index of the module is at least 2. All of the above results are closely tied to the cancellation problem for modules over commutative rings and this was the topic of Andrei’s talk at ICM 1978 in Helsinki. It would be the first of 3 ICM talks he was invited to give, the second of which would be a plenary talk. In the years immediately after the Helsinki meeting, his work included the normality of the elementary subgroup of a general linear group of rank at least 3 over a module finite ring (one which is finitely generated as a module over its center), the centrality of any Steinberg group extension of rank at least 4 over a module finite ring (joint with his student M.S. Tulenbaev), and his stability theorem for higher K-groups over rings of finite stable rank. The first and last paper of the current issue use ideas, techniques, and results from the early period above. At the beginning of the 1980s, Andrei turned his attention to the K-theory of fields and division rings. By a classical result of Kummer, the norm residue homomorphism of degree 1 was known to be an isomorphism. In 1982, Andrei showed together with A. Merkurjev that the norm residue homomorphism of degree","PeriodicalId":50167,"journal":{"name":"Journal of K-Theory","volume":"83 6 1","pages":"403-405"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89312682","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 : 2010-06-01DOI: 10.1017/IS010005001JKT101
Selby Jose, R. A. Rao
We describe a homomorphism from the group SUmr (R), generated by Suslin matrices, when r is even, to the special orthogonal group SO2(r+1) (R) by relating the Suslin matrix corresponding to a pair of vectors v, w, with 〈v, w〉 = 1, to the product of two reflections, one w.r.t. the vectors v, w and the other w.r.t. the vectors e1, e1 (of length one). When r is odd we can still associate a product of reflections with an element of SUmr (R), which is well defined up to a unit u, with u2 = 1. This association enables one to study the orbit space of unimodular vectors under the elementary subgroup.
当R为偶时，由Suslin矩阵生成的群SUmr (R)与特殊正交群SO2(R +1) (R)的同态，通过将Suslin矩阵对应于一对向量v, w， < v, w > = 1，与两个反射的乘积联系起来，一个是向量v, w，另一个是向量e1, e1(长度为1)。当r为奇数时，我们仍然可以将反射的乘积与SUmr (r)中的一个元素联系起来，这个元素定义得很好，直到单位u, u2 = 1。这种联系使人们能够研究初等子群下的单模向量的轨道空间。
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Pub Date : 2010-06-01DOI: 10.1017/is010004028jkt103
I. Fesenko
We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K 2 -delic and K 1 × K 1 -delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.
构造了算术曲面上秩二积分结构的积分对象，发展了测度和积分理论，以及谐波分析的基本原理。利用与算术曲面相关的拓扑Milnor K 2 -delic和K 1 × k1 -delic对象，定义了一个共轭ζ积分。它的非分支形式与表面的zeta函数的平方密切相关。对于全局场上椭圆曲线的适当正则模型，导出了Tate和Iwasawa理论的二维版本。利用阿德利奇解析对偶性和二维公式，将对ζ积分的研究简化为对边界积分项的研究。本文首先应用了zeta函数的三个基本性质:它的亚纯延拓和泛函方程，以及关于它的平均周期的假设;它的极点的位置和关于边界函数的四阶对数导数的符号的持久性的假设它的极点是边界积分的中心点在这一点上，边界积分明确地联系了解析秩和算术秩。
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Pub Date : 2010-06-01DOI: 10.1017/IS010003002JKT111
R. Hazrat, V. Petrov, N. Vavilov
We finish the proof of the main structure theorems for a Chevalley group G (Φ, R ) of rank ≥ 2 over an arbitrary commutative ring R . Namely, we prove that for any admissible pair ( A, B ) in the sense of Abe, the corresponding relative elementary group E (Φ, R, A, B ) and the full congruence subgroup C (Φ, R, A, B ) are normal in G (Φ, R ) itself, and not just normalised by the elementary group E (Φ, R ) and that [ E (Φ, R ), C (Φ, R, A, B )] = E , (Φ, R, A, B ). For the case Φ = F 4 these results are new. The proof is new also for other cases, since we explicitly define C (Φ, R, A, B ) by congruences in the adjoint representation of G (Φ, R ) and give several equivalent characterisations of that group and use these characterisations in our proof.
我们完成了秩≥2的Chevalley群G (Φ， R)在任意交换环R上的主要结构定理的证明。也就是说，我们证明了对于Abe意义上的任何可容许对(A, B)，相应的相对初等群E (Φ， R, A, B)和满同余子群C (Φ， R, A, B)在G (Φ， R)本身中是正规的，而不仅仅是被初等群E (Φ， R)规整，并且[E (Φ， R)， C (Φ， R, A, B)] = E， (Φ， R, A, B)。对于Φ = f4的情况，这些结果是新的。这个证明对于其他情况也是新的，因为我们通过G (Φ， R)的伴随表示中的同余显式地定义了C (Φ， R, A, B)，并给出了该群的几个等价特征，并在我们的证明中使用了这些特征。
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Pub Date : 2010-04-01DOI: 10.1017/IS010001014JKT096
Matthias Wendt
In this paper, we describe the sheaves of A-homotopy groups of a simplyconnected Chevalley group G. The A-homotopy group sheaves can be identified with the sheafification of the unstable Karoubi-Villamayor K-groups.
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Pub Date : 2010-02-28DOI: 10.1017/is010008010jkt126
Ivo Dell’Ambrogio
Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category Boot ⊂ KK of separable complex C*-algebras. Namely, they are in a bijective correspondence with subsets of the Zariski spectrum Specℤ of the integers – precisely as for the localizing subcategories of the derived category D(ℤ) of complexes of abelian groups. We provide corollaries of this fact and put it in context with the similar classifications available in the literature.
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