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Stable operations and cooperations in derived Witt theory with rational coefficients 具有有理系数的推导Witt理论中的稳定操作与合作
arXiv: K-Theory and Homology Pub Date : 2015-04-19 DOI: 10.2140/akt.2017.2.517
A. Ananyevskiy
{"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}
引用次数: 11
Comparison of KE-Theory and KK-Theory ke理论与kk理论的比较
arXiv: K-Theory and Homology Pub Date : 2015-03-13 DOI: 10.4171/JNCG/256
R. Meyer
{"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}
引用次数: 0
Homological algebra for commutative monoids 交换模群的同调代数
arXiv: K-Theory and Homology Pub Date : 2015-03-08 DOI: 10.7282/T3N58P2X
J. Flores
{"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}
引用次数: 6
A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities 具有系数和Kleinian奇点的代数k理论的a1 -同伦不变性
arXiv: K-Theory and Homology Pub Date : 2015-02-18 DOI: 10.2140/AKT.2017.2.1
Gonçalo Tabuada
{"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}
引用次数: 13
The extension class and KMS states for Cuntz--Pimsner algebras of some bi-Hilbertian bimodules 某些双模的Cuntz—Pimsner代数的扩展类和KMS态
arXiv: K-Theory and Homology Pub Date : 2015-01-22 DOI: 10.1142/S1793525317500108
A. Rennie, D. Robertson, A. Sims
{"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}
引用次数: 25
Multiplicative Structures and the Twisted Baum-Connes Assembly map 乘法结构和扭曲的Baum-Connes组装图
arXiv: K-Theory and Homology Pub Date : 2015-01-21 DOI: 10.1090/TRAN/7024
No'e B'arcenas, P. C. Rouse, Mario Vel'asquez
{"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}
引用次数: 5
Splitting the relative assembly map, Nil-terms and involutions 拆分相对组装图、零项和内联
arXiv: K-Theory and Homology Pub Date : 2015-01-12 DOI: 10.2140/akt.2016.1.339
W. Lueck, W. Steimle
{"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}
引用次数: 12
On localization sequences in the algebraic K-theory of ring spectra 环谱代数k理论中的局部化序列
arXiv: K-Theory and Homology Pub Date : 2014-12-12 DOI: 10.4171/JEMS/771
Benjamin Antieau, T. Barthel, David Gepner
{"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}
引用次数: 9
Coarse co-assembly as a ring homomorphism 作为环同态的粗共装配
arXiv: K-Theory and Homology Pub Date : 2014-12-04 DOI: 10.4171/JNCG/240
Christopher Wulff
{"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}
引用次数: 6
A categorification of the square root of -1 -1的平方根的分类
arXiv: K-Theory and Homology Pub Date : 2014-11-27 DOI: 10.4064/FM232-1-7
Yin Tian
{"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}
引用次数: 1
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