The algebras of stable operations and cooperations in derived Witt theory with rational coefficients are computed and an additive description of cooperations in derived Witt theory is given. The answer is parallel to the well-known case of K-theory of real vector bundles in topology. In particular, we show that stable operations in derived Witt theory with rational coefficients are given by the values on the powers of Bott element.
{"title":"Stable operations and cooperations in derived Witt theory with rational coefficients","authors":"A. Ananyevskiy","doi":"10.2140/akt.2017.2.517","DOIUrl":"https://doi.org/10.2140/akt.2017.2.517","url":null,"abstract":"The algebras of stable operations and cooperations in derived Witt theory with rational coefficients are computed and an additive description of cooperations in derived Witt theory is given. The answer is parallel to the well-known case of K-theory of real vector bundles in topology. In particular, we show that stable operations in derived Witt theory with rational coefficients are given by the values on the powers of Bott element.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133700011","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 the character from the bivariant K-theory KE^G introduced by Dumitrascu to E^G factors through Kasparov's KK^G for any locally compact group G. Hence KE^G contains KK^G as a direct summand.
{"title":"Comparison of KE-Theory and KK-Theory","authors":"R. Meyer","doi":"10.4171/JNCG/256","DOIUrl":"https://doi.org/10.4171/JNCG/256","url":null,"abstract":"We show that the character from the bivariant K-theory KE^G introduced by Dumitrascu to E^G factors through Kasparov's KK^G for any locally compact group G. Hence KE^G contains KK^G as a direct summand.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131370485","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 first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. After giving the definition of, and basic results for, $A$-sets, we classify projective $A$-sets and show they are completely determine by their rank. Subsequently, for a monoid $A$, we compute $K_0$ and $K_1$ and prove the Devissage Theorem for $G_0$. With the definition of short exact sequence for $A$-sets in hand, we describe the set $Ext(X,Y)$ of extensions for $A$-sets $X,Y$ and classify the set of square-zero extensions of a monoid $A$ by an $A$-set $X$ using the Hochschild cosimplicial set. We also examine the projective model structure on simplicial $A$-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category $operatorname{Da}(mathcal{C})$ of double-arrow complexes for a class of non-abelian categories $mathcal{C}$ and, in the case of $A$-sets, shows an adjunction with the category of simplicial $A$-sets.
我们首先研究可交换的点模群，以类似于可交换环理论的方式给出了基本的定义和结果。给出了一类特殊单群的链条件、初等分解和归一化的结果，从而研究了一类单群方案、除数、Picard群和类群。证明了一元的归一化不一定是一元，但可能是一元格式。在给出了$A$集的定义和基本结果之后，我们对投影$A$集进行了分类，并证明它们完全由秩决定。随后，我们对一元$ a $计算了$K_0$和$K_1$，并证明了$G_0$的设计定理。有了A$-集合的短精确序列的定义，我们描述了A$-集合X,Y$的扩展集$Ext(X,Y)$，并利用Hochschild协简集对一元$A$的平方零扩展集$A$进行分类。我们还研究了简单$A$-集上的投影模型结构，展示了计算同伦群以及确定单群的派生范畴所涉及的困难。对于一类非阿贝尔范畴$mathcal{C}$，定义了双箭头复形的范畴$operatorname{Da}(mathcal{C})$，并在$ a $-sets的情况下，给出了与简单$ a $-sets范畴的一个附加关系。
{"title":"Homological algebra for commutative monoids","authors":"J. Flores","doi":"10.7282/T3N58P2X","DOIUrl":"https://doi.org/10.7282/T3N58P2X","url":null,"abstract":"We first study commutative, pointed monoids providing basic definitions and results in a manner similar commutative ring theory. Included are results on chain conditions, primary decomposition as well as normalization for a special class of monoids which lead to a study monoid schemes, divisors, Picard groups and class groups. It is shown that the normalization of a monoid need not be a monoid, but possibly a monoid scheme. \u0000After giving the definition of, and basic results for, $A$-sets, we classify projective $A$-sets and show they are completely determine by their rank. Subsequently, for a monoid $A$, we compute $K_0$ and $K_1$ and prove the Devissage Theorem for $G_0$. With the definition of short exact sequence for $A$-sets in hand, we describe the set $Ext(X,Y)$ of extensions for $A$-sets $X,Y$ and classify the set of square-zero extensions of a monoid $A$ by an $A$-set $X$ using the Hochschild cosimplicial set. \u0000We also examine the projective model structure on simplicial $A$-sets showcasing the difficulties involved in computing homotopy groups as well as determining the derived category for a monoid. The author defines the category $operatorname{Da}(mathcal{C})$ of double-arrow complexes for a class of non-abelian categories $mathcal{C}$ and, in the case of $A$-sets, shows an adjunction with the category of simplicial $A$-sets.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126215281","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}
C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the Kleinian singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain some vanishing and divisibility properties of algebraic K-theory (without coefficients).
C. Weibel和Thomason-Trobaugh(在某些假设下)证明了带系数的代数k理论是a1 -同伦不变的。在本文中，我们将这一结果从方案推广到dg范畴的广义集。在此过程中，我们将Bass-Quillen的基本定理以及Stienstra关于大威特环上模结构的基础工作扩展到dg范畴的集合。在其他情况下，上面的a1 -同伦不变性结果现在可以应用于堆栈上的dg代数(不一定是交换的)。作为一个应用，我们仅利用Coxeter矩阵的核和核，计算了dg类范畴系数的代数k理论。这导致了一个完整的代数k理论的计算，其中Kleinian奇点的系数由简单的带条纹的Dynkin图参数化。作为一个副产品，我们得到了代数k理论(无系数)的一些消失性和可整除性。
{"title":"A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities","authors":"Gonçalo Tabuada","doi":"10.2140/AKT.2017.2.1","DOIUrl":"https://doi.org/10.2140/AKT.2017.2.1","url":null,"abstract":"C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the Kleinian singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain some vanishing and divisibility properties of algebraic K-theory (without coefficients).","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130349815","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 : 2015-01-22DOI: 10.1142/S1793525317500108
A. Rennie, D. Robertson, A. Sims
For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara--Pinzari--Watatani, we construct a Kasparov module representing the extension class defining the Cuntz--Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^*$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz--Pimsner algebras.
{"title":"The extension class and KMS states for Cuntz--Pimsner algebras of some bi-Hilbertian bimodules","authors":"A. Rennie, D. Robertson, A. Sims","doi":"10.1142/S1793525317500108","DOIUrl":"https://doi.org/10.1142/S1793525317500108","url":null,"abstract":"For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara--Pinzari--Watatani, we construct a Kasparov module representing the extension class defining the Cuntz--Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^*$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz--Pimsner algebras.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125391498","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}
Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.
{"title":"Multiplicative Structures and the Twisted Baum-Connes Assembly map","authors":"No'e B'arcenas, P. C. Rouse, Mario Vel'asquez","doi":"10.1090/TRAN/7024","DOIUrl":"https://doi.org/10.1090/TRAN/7024","url":null,"abstract":"Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-theory groups whenever this assembly maps are isomorphisms.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124345136","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 the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.
{"title":"Splitting the relative assembly map, Nil-terms and involutions","authors":"W. Lueck, W. Steimle","doi":"10.2140/akt.2016.1.339","DOIUrl":"https://doi.org/10.2140/akt.2016.1.339","url":null,"abstract":"We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115224847","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 identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a negative answer to a question of Rognes for $n>1$ by comparing the traces of the fiber of the map $K(BP(n))rightarrow K(E(n))$ and of $K(BP(n-1))$ in rational topological Hochschild homology.
{"title":"On localization sequences in the algebraic K-theory of ring spectra","authors":"Benjamin Antieau, T. Barthel, David Gepner","doi":"10.4171/JEMS/771","DOIUrl":"https://doi.org/10.4171/JEMS/771","url":null,"abstract":"We identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a negative answer to a question of Rognes for $n>1$ by comparing the traces of the fiber of the map $K(BP(n))rightarrow K(E(n))$ and of $K(BP(n-1))$ in rational topological Hochschild homology.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125058392","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 $K$-theory of the stable Higson corona of a coarse space carries a canonical ring structure. This ring is the domain of an unreduced version of the coarse co-assembly map of Emerson and Meyer. We show that the target also carries a ring structure and co-assembly is a ring homomorphism, provided that the given coarse space is contractible in a coarse sense.
{"title":"Coarse co-assembly as a ring homomorphism","authors":"Christopher Wulff","doi":"10.4171/JNCG/240","DOIUrl":"https://doi.org/10.4171/JNCG/240","url":null,"abstract":"The $K$-theory of the stable Higson corona of a coarse space carries a canonical ring structure. This ring is the domain of an unreduced version of the coarse co-assembly map of Emerson and Meyer. We show that the target also carries a ring structure and co-assembly is a ring homomorphism, provided that the given coarse space is contractible in a coarse sense.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116628435","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 give a graphical calculus for a monoidal DG category $cal{I}$ whose Grothendieck group is isomorphic to the ring $mathbb{Z}[sqrt{-1}]$. We construct a categorical action of $cal{I}$ which lifts the action of $mathbb{Z}[sqrt{-1}]$ on $mathbb{Z}^2$.
{"title":"A categorification of the square root of -1","authors":"Yin Tian","doi":"10.4064/FM232-1-7","DOIUrl":"https://doi.org/10.4064/FM232-1-7","url":null,"abstract":"We give a graphical calculus for a monoidal DG category $cal{I}$ whose Grothendieck group is isomorphic to the ring $mathbb{Z}[sqrt{-1}]$. We construct a categorical action of $cal{I}$ which lifts the action of $mathbb{Z}[sqrt{-1}]$ on $mathbb{Z}^2$.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132421096","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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