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Groups with Spanier-Whitehead Duality 具有西班牙-怀特黑德二元性的群体
arXiv: K-Theory and Homology Pub Date : 2019-08-10 DOI: 10.14760/OWP-2019-23
Shintaro Nishikawa, Valerio Proietti
{"title":"Groups with Spanier-Whitehead Duality","authors":"Shintaro Nishikawa, Valerio Proietti","doi":"10.14760/OWP-2019-23","DOIUrl":"https://doi.org/10.14760/OWP-2019-23","url":null,"abstract":"We introduce the notion of Spanier-Whitehead K-duality for a discrete group G, defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard \"gamma element\" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115161500","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}
引用次数: 6
Homotopy equivalence in unboundedKK-theory 无界kk理论中的同伦等价
arXiv: K-Theory and Homology Pub Date : 2019-07-09 DOI: 10.2140/AKT.2020.5.501
Koen van den Dungen, B. Mesland
{"title":"Homotopy equivalence in unbounded\u0000KK-theory","authors":"Koen van den Dungen, B. Mesland","doi":"10.2140/AKT.2020.5.501","DOIUrl":"https://doi.org/10.2140/AKT.2020.5.501","url":null,"abstract":"We propose a new notion of unbounded $K!K$-cycle, mildly generalising unbounded Kasparov modules, for which the direct sum is well-defined. To a pair $(A,B)$ of $sigma$-unital $C^{*}$-algebras, we can then associate a semigroup $overline{U!K!K}(A,B)$ of homotopy equivalence classes of unbounded cycles, and we prove that this semigroup is in fact an abelian group. In case $A$ is separable, our group $overline{U!K!K}(A,B)$ is isomorphic to Kasparov's $K!K$-theory group $K!K(A,B)$ via the bounded transform. We also discuss various notions of degenerate cycles, and we prove that the homotopy relation on unbounded cycles coincides with the relation generated by operator-homotopies and addition of degenerate cycles.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128479643","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}
引用次数: 7
On the classification of group actions on C*-algebras up to equivariant KK-equivalence C*-代数上直至等变kk等价的群作用的分类
arXiv: K-Theory and Homology Pub Date : 2019-06-26 DOI: 10.2140/akt.2021.6.157
R. Meyer
{"title":"On the classification of group actions on C*-algebras up to equivariant KK-equivalence","authors":"R. Meyer","doi":"10.2140/akt.2021.6.157","DOIUrl":"https://doi.org/10.2140/akt.2021.6.157","url":null,"abstract":"We study the classification of group actions on C*-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite C*-algebra. We show that a conjecture of Izumi is equivalent to an equivalence between cocycle conjugacy and equivariant KK-equivalence for actions of torsion-free amenable groups on Kirchberg algebras. Let G be a cyclic group of prime order. We describe its actions up to equivariant KK-equivalence, based on previous work by Manuel Kohler. In particular, we classify actions of G on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131963200","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}
引用次数: 13
The tangent complex of K-theory k理论的正切复合体
arXiv: K-Theory and Homology Pub Date : 2019-04-17 DOI: 10.5802/JEP.161
Benjamin Hennion
{"title":"The tangent complex of K-theory","authors":"Benjamin Hennion","doi":"10.5802/JEP.161","DOIUrl":"https://doi.org/10.5802/JEP.161","url":null,"abstract":"We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. \u0000We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_infty to K$. \u0000The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121045085","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}
引用次数: 0
Crossed modules and symmetric cohomology of groups 群的交叉模与对称上同调
arXiv: K-Theory and Homology Pub Date : 2019-02-05 DOI: 10.4310/hha.2020.v22.n2.a7
Mariam Pirashvili
{"title":"Crossed modules and symmetric cohomology of groups","authors":"Mariam Pirashvili","doi":"10.4310/hha.2020.v22.n2.a7","DOIUrl":"https://doi.org/10.4310/hha.2020.v22.n2.a7","url":null,"abstract":"This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126711905","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}
引用次数: 3
Regularity of Spectral Stacks and Discreteness of Weight-Hearts 谱叠加的正则性与权重心的离散性
arXiv: K-Theory and Homology Pub Date : 2019-01-08 DOI: 10.1093/QMATH/HAAB017
Adeel A. Khan, V. Sosnilo
{"title":"Regularity of Spectral Stacks and Discreteness of Weight-Hearts","authors":"Adeel A. Khan, V. Sosnilo","doi":"10.1093/QMATH/HAAB017","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB017","url":null,"abstract":"We study regularity in the context of ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[operatorname{Spec} R/G]$ defined over a field, where $R$ is a noetherian ${mathcal{E}_{infty}}$-$k$-algebra and $G$ is a linearly reductive group acting on $R$. In this context we show that regularity of $X$ is equivalent to regularity of $R$. We also show that if $R$ is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. \u0000We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $infty$-categories.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121888334","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}
引用次数: 8
Tannaka duality for enhanced triangulated categories I: reconstruction 增强三角分类I的Tannaka对偶性:重建
arXiv: K-Theory and Homology Pub Date : 2018-12-27 DOI: 10.4171/jncg/374
J. Pridham
{"title":"Tannaka duality for enhanced triangulated categories I: reconstruction","authors":"J. Pridham","doi":"10.4171/jncg/374","DOIUrl":"https://doi.org/10.4171/jncg/374","url":null,"abstract":"We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $mathcal{A}$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $mathcal{A}$-modules and of $C$-comodules. When $mathcal{A}$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116278494","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}
引用次数: 1
The Higson–Roe sequence for étale groupoids. II. The universal sequence for equivariant families 杂群的Higson-Roe序列。2等变族的普遍序列
arXiv: K-Theory and Homology Pub Date : 2018-12-11 DOI: 10.4171/JNCG/394
M. Benameur, Indrava Roy
{"title":"The Higson–Roe sequence for étale groupoids. II. The universal sequence for equivariant families","authors":"M. Benameur, Indrava Roy","doi":"10.4171/JNCG/394","DOIUrl":"https://doi.org/10.4171/JNCG/394","url":null,"abstract":"This is the second part of our series about the Higson-Roe sequence for etale groupoids. We devote this part to the proof of the universal $K$-theory surgery exact sequence which extends the seminal results of N. Higson and J. Roe to the case of transformation groupoids. In the process, we prove the expected functoriality properties as well as the Paschke-Higson duality theorem.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128860245","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}
引用次数: 5
Periodicity and cyclic homology. Para-$S$-modules and perturbation lemmas 周期性和循环同调。Para-$S -模与摄动引理
arXiv: K-Theory and Homology Pub Date : 2018-10-11 DOI: 10.4171/jncg/393
Raphael Ponge
{"title":"Periodicity and cyclic homology. Para-$S$-modules and perturbation lemmas","authors":"Raphael Ponge","doi":"10.4171/jncg/393","DOIUrl":"https://doi.org/10.4171/jncg/393","url":null,"abstract":"In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para-$S$-module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. We establish some generalizations for para-$S$-modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hoschschild chain complexes into deformation retracts of para-$S$-modules. By using ideas of Kassel this then allows us to get comparison results between the various para-$S$-modules associated with para-precyclic modules, and between them and Connes' cyclic chain complex. These comparison results lead us to alternative descriptions of Connes' periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic $(b,B)$-cocycle into a cohomologous periodic cyclic cocycle.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131513022","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}
引用次数: 0
On the Hochschild homology of $ell^1$-rapid decay group algebras 关于$ell^1$-快速衰变群代数的Hochschild同调
arXiv: K-Theory and Homology Pub Date : 2018-09-04 DOI: 10.4171/ggd/586
A. Engel
{"title":"On the Hochschild homology of $ell^1$-rapid decay group algebras","authors":"A. Engel","doi":"10.4171/ggd/586","DOIUrl":"https://doi.org/10.4171/ggd/586","url":null,"abstract":"We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the l^1-rapid decay group algebra is injective.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131768648","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}
引用次数: 0
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