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.
{"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}
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.
{"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}
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.
{"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}
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. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_infty to K$. The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.
我们证明了在特征0场k上的(阿贝尔)变形问题中，k理论的切复是循环同调的(在k上)。这个等价与$lambda$ -运算相容。特别是，相对代数k -理论函子完全确定了特征为0的任意域k上的绝对循环同调。我们还证明Loday-Quillen-Tsygan广义迹是规范映射$BGL_infty to K$的切态射。这个证明建立在Goodwillie的结果之上，使用了Wodzicki对循环同调的剔除和Lurie-Pridham的形式变形理论。
{"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}
Pub Date : 2019-02-05DOI: 10.4310/hha.2020.v22.n2.a7
Mariam Pirashvili
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.
{"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}
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. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $infty$-categories.
{"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}
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.
{"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}
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.
{"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}
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.
在本文中，我们引入了$S$-模块的一个副环版本。这些新对象被称为para-$S -模块。副环模和副链配合物产生对S -模的方式与循环模和混合配合物产生S -模的方式非常相似。更一般地说，para- S -模为我们提供了一个自然的框架来得到各种结构的副环模和副链配合物的类似物以及环模或混合配合物的等价结果。para-$S -模的基准不提供链复形，因此同构和拟同构的概念没有意义。建立了微分同调代数基本摄动引理的对- S -模和对链复的一些推广。这些推广为我们提供了将Hoschschild链配合物的变形缩回转化为对S模的变形缩回的一般方法。通过使用Kassel的思想，我们可以得到与对预环模相关的各种对预环模之间的比较结果，以及它们与Connes环链复合物之间的比较结果。这些比较结果使我们得到了对Connes周期算子的不同描述。这在周期循环同调中有一些应用。我们还描述了这些结果在循环上同中的对应关系。特别地，我们得到了将周期$(b, b)$-环转化为上同源周期循环环的显式方法。
{"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}
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.
我们证明了对于许多半双曲群，l^1快速衰减群代数的Hochschild同调的共轭类分解是内射的。
{"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}
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