Pub Date : 2015-11-18DOI: 10.4310/HHA.2017.V19.N1.A17
O. Braunling, M. Groechenig, J. Wolfson
We investigate the properties of relative analogues of admissible Ind, Pro, and elementary Tate objects for pairs of exact categories, and give criteria for those categories to be abelian. A relative index map is introduced, and as an application we deduce a description for boundary morphisms in the K-theory of coherent sheaves on Noetherian schemes.
{"title":"Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves","authors":"O. Braunling, M. Groechenig, J. Wolfson","doi":"10.4310/HHA.2017.V19.N1.A17","DOIUrl":"https://doi.org/10.4310/HHA.2017.V19.N1.A17","url":null,"abstract":"We investigate the properties of relative analogues of admissible Ind, Pro, and elementary Tate objects for pairs of exact categories, and give criteria for those categories to be abelian. A relative index map is introduced, and as an application we deduce a description for boundary morphisms in the K-theory of coherent sheaves on Noetherian schemes.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130825238","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 : 2015-11-09DOI: 10.1007/978-3-319-43674-6_2
D. Juan-Pineda, Luis Jorge S'anchez Saldana
{"title":"The K and L Theoretic Farrell-Jones Isomorphism Conjecture for Braid Groups","authors":"D. Juan-Pineda, Luis Jorge S'anchez Saldana","doi":"10.1007/978-3-319-43674-6_2","DOIUrl":"https://doi.org/10.1007/978-3-319-43674-6_2","url":null,"abstract":"","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130049791","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 consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.
{"title":"A Baum–Connes conjecture for singular\u0000foliations","authors":"Iakovos Androulidakis, G. Skandalis","doi":"10.2140/akt.2019.4.561","DOIUrl":"https://doi.org/10.2140/akt.2019.4.561","url":null,"abstract":"We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132670428","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}
The main goal of this paper is to prove the following: for a triangulated category $ underline{C}$ and $Esubset operatorname{Obj} underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that $E$ is its set of zeros if (and only if) $E$ is closed with respect to retracts and extensions (so, we obtain a certain Nullstellensatz for functors of this type). Moreover, for $ underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: underline{C}^{op}to R-operatorname{mod}$ satisfying this property. As a corollary, we prove that an object $Y$ belongs to the corresponding "envelope" of some $Dsubset operatorname{Obj} underline{C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $ underline{C}_p$ obtained from $ underline{C}$ by means of "localizing the coefficients" at maximal ideals $ptriangleleft R$. Moreover, to prove our theorem we develop certain new methods for relating triangulated categories to their (non-full) countable triangulated subcategories. The results of this paper can be applied to the study of weight structures and of triangulated categories of motives.
{"title":"A Nullstellensatz for triangulated categories","authors":"M. Bondarko, V. Sosnilo","doi":"10.1090/SPMJ/1425","DOIUrl":"https://doi.org/10.1090/SPMJ/1425","url":null,"abstract":"The main goal of this paper is to prove the following: for a triangulated category $ underline{C}$ and $Esubset operatorname{Obj} underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that $E$ is its set of zeros if (and only if) $E$ is closed with respect to retracts and extensions (so, we obtain a certain Nullstellensatz for functors of this type). Moreover, for $ underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: underline{C}^{op}to R-operatorname{mod}$ satisfying this property. \u0000As a corollary, we prove that an object $Y$ belongs to the corresponding \"envelope\" of some $Dsubset operatorname{Obj} underline{C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $ underline{C}_p$ obtained from $ underline{C}$ by means of \"localizing the coefficients\" at maximal ideals $ptriangleleft R$. Moreover, to prove our theorem we develop certain new methods for relating triangulated categories to their (non-full) countable triangulated subcategories. \u0000The results of this paper can be applied to the study of weight structures and of triangulated categories of motives.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126558660","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 note we extend the classical result by G. G. Kasparov that the Kasparov groups $KK_1(A,B)$ can be identified with the extension groups $mbox{Ext}(A,B)$ to the inverse semigroup equivariant setting. More precisely, we show that $KK_G^1(A,B) cong mbox{Ext}_G(A otimes {cal K}_G,B otimes {cal K}_G)$ for every countable, $E$-continuous inverse semigroup $G$. For locally compact second countable groups $G$ this was proved by K. Thomsen, and technically this note presents an adaption of his proof.
{"title":"Inverse semigroup equivariant $KK$-theory and $C^*$-extensions","authors":"B. Burgstaller","doi":"10.7153/OAM-10-27","DOIUrl":"https://doi.org/10.7153/OAM-10-27","url":null,"abstract":"In this note we extend the classical result by G. G. Kasparov that the Kasparov groups $KK_1(A,B)$ can be identified with the extension groups $mbox{Ext}(A,B)$ to the inverse semigroup equivariant setting. More precisely, we show that $KK_G^1(A,B) cong mbox{Ext}_G(A otimes {cal K}_G,B otimes {cal K}_G)$ for every countable, $E$-continuous inverse semigroup $G$. For locally compact second countable groups $G$ this was proved by K. Thomsen, and technically this note presents an adaption of his proof.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131050252","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 : 2015-07-06DOI: 10.1142/S0219498818500913
J. Guccione, J. Guccione, C. Valqui
Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to ker(p). This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.
{"title":"Cyclic homology of cleft extensions of algebras","authors":"J. Guccione, J. Guccione, C. Valqui","doi":"10.1142/S0219498818500913","DOIUrl":"https://doi.org/10.1142/S0219498818500913","url":null,"abstract":"Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to ker(p). This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116097958","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 : 2015-06-29DOI: 10.1142/S1005386717000128
R. Preusser
This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,Lambda)$ which are normalized by the elementary subgroup $EU_{2n}(R,Lambda)$, under the condition that $R$ is a quasi-finite ring with involution, i.e a direct limit of module finite rings with involution, and $ngeq 3$.
{"title":"Structure of Hyperbolic Unitary Groups II: Classification of E-normal Subgroups","authors":"R. Preusser","doi":"10.1142/S1005386717000128","DOIUrl":"https://doi.org/10.1142/S1005386717000128","url":null,"abstract":"This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,Lambda)$ which are normalized by the elementary subgroup $EU_{2n}(R,Lambda)$, under the condition that $R$ is a quasi-finite ring with involution, i.e a direct limit of module finite rings with involution, and $ngeq 3$.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115960684","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 establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.
{"title":"On the equivariant K-homology of PSL_2 of the imaginary quadratic integers","authors":"Alexander D. Rahm","doi":"10.5802/AIF.3047","DOIUrl":"https://doi.org/10.5802/AIF.3047","url":null,"abstract":"We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129111952","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 Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injective properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to non-torsion cohomology classes and dg Azumaya algebras associated to torsion cohomology classes (=ordinary Azumaya algebras); in the case of a field of characteristic zero, it is injective; in the case of a field of positive characteristic p>0, it restricts to an injective map on the p-primary component of the Brauer group.
{"title":"A note on secondary K-theory","authors":"Gonçalo Tabuada","doi":"10.2140/ANT.2016.10.887","DOIUrl":"https://doi.org/10.2140/ANT.2016.10.887","url":null,"abstract":"We prove that Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injective properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to non-torsion cohomology classes and dg Azumaya algebras associated to torsion cohomology classes (=ordinary Azumaya algebras); in the case of a field of characteristic zero, it is injective; in the case of a field of positive characteristic p>0, it restricts to an injective map on the p-primary component of the Brauer group.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120882876","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 : 2015-05-13DOI: 10.2140/agt.2016.16.2107
Mauricio Bustamante, Luis Jorge S'anchez Saldana
We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.
{"title":"On the algebraic $K$-theory of the Hilbert modular group","authors":"Mauricio Bustamante, Luis Jorge S'anchez Saldana","doi":"10.2140/agt.2016.16.2107","DOIUrl":"https://doi.org/10.2140/agt.2016.16.2107","url":null,"abstract":"We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122960068","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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