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Batalin-Vilkovisky structure over the Hochschild cohomology ring of a group algebra 群代数的Hochschild上同调环上的Batalin-Vilkovisky结构
arXiv: K-Theory and Homology Pub Date : 2014-05-14 DOI: 10.4171/JNCG/249
Yuming Liu, Guodong Zhou
{"title":"Batalin-Vilkovisky structure over the Hochschild cohomology ring of a group algebra","authors":"Yuming Liu, Guodong Zhou","doi":"10.4171/JNCG/249","DOIUrl":"https://doi.org/10.4171/JNCG/249","url":null,"abstract":"We realize explicitly the well-known additive decomposition of the Hochschild cohomology ring of a group algebra in the elements level. As a result, we describe the cup product, the Batalin-Vilkovisky operator and the Lie bracket in the Hochschild cohomology ring of a group algebra.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132841110","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: 16
Cohomology of presheaves with oriented weak transfers 具有定向弱转移的预轴的上同性
arXiv: K-Theory and Homology Pub Date : 2014-05-01 DOI: 10.1090/JAG/684
J. Ross
{"title":"Cohomology of presheaves with oriented weak transfers","authors":"J. Ross","doi":"10.1090/JAG/684","DOIUrl":"https://doi.org/10.1090/JAG/684","url":null,"abstract":"Over a field of characteristic zero, we establish the homotopy invariance of the Nisnevich cohomology of homotopy invariant presheaves with oriented weak transfers, and the agreement of Zariski and Nisnevich cohomology for such presheaves. This generalizes a foundational result in Voevodsky's theory of motives. The main idea is to find explicit smooth representatives of the correspondences which provide the input for Voevodsky's cohomological architecture.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121915932","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
Random graphs, weak coarse embeddings, and higher index theory 随机图，弱粗嵌入，高指标理论
arXiv: K-Theory and Homology Pub Date : 2014-04-25 DOI: 10.1142/S1793525315500156
R. Willett
{"title":"Random graphs, weak coarse embeddings, and higher index theory","authors":"R. Willett","doi":"10.1142/S1793525315500156","DOIUrl":"https://doi.org/10.1142/S1793525315500156","url":null,"abstract":"This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. \u0000The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132333019","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
Pro unitality and pro excision in algebraic K-theory and cyclic homology 代数k理论与循环同调中的亲酉性与亲切性
arXiv: K-Theory and Homology Pub Date : 2014-04-16 DOI: 10.1515/CRELLE-2015-0007
M. Morrow
{"title":"Pro unitality and pro excision in algebraic K-theory and cyclic homology","authors":"M. Morrow","doi":"10.1515/CRELLE-2015-0007","DOIUrl":"https://doi.org/10.1515/CRELLE-2015-0007","url":null,"abstract":"We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Corti~nas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of Andr'e--Quillen, Hochschild, and cyclic homology. Our key tool is to first establish the equivalence of various pro Tor vanishing conditions which appear in the literature. Using this we prove that all ideals of commutative, Noetherian rings are pro unital in a certain sense, and show that such ideals satisfy pro excision in $K$-theory as well as in cyclic and topological cyclic homology. In addition, our techniques yield a strong form of the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a generalisation of the Feu{i}gin--Tsygan theorem.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114366877","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: 29
A relative version of the Beilinson-Hodge conjecture 贝林森-霍奇猜想的一个相对版本
arXiv: K-Theory and Homology Pub Date : 2014-04-03 DOI: 10.1017/CBO9781316387887.010
Rob de Jeu, James D. Lewis, D. Patel
{"title":"A relative version of the Beilinson-Hodge conjecture","authors":"Rob de Jeu, James D. Lewis, D. Patel","doi":"10.1017/CBO9781316387887.010","DOIUrl":"https://doi.org/10.1017/CBO9781316387887.010","url":null,"abstract":"Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127708233","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: 2
Revisiting Farrell’s nonfiniteness of Nil 重新审视法雷尔的零的非有限性
arXiv: K-Theory and Homology Pub Date : 2014-03-27 DOI: 10.2140/akt.2016.1.209
J.-F. Lafont, S. Prassidis, Kun Wang
{"title":"Revisiting Farrell’s nonfiniteness of Nil","authors":"J.-F. Lafont, S. Prassidis, Kun Wang","doi":"10.2140/akt.2016.1.209","DOIUrl":"https://doi.org/10.2140/akt.2016.1.209","url":null,"abstract":"We study Farrell Nil-groups associated to a finite order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. We use this to deduce a structure theorem for countable Farrell Nil-groups with finite exponent. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group, provided they have finite exponent (which is always the case in dimension $0$).","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133571104","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
Cohomology of $mathcal A_theta^{alg} rtimes mathbb Z_2$ and its Chern-Connes pairing $mathcal A_theta^{alg} r_乘以mathbb Z_2$的上同调及其chen - connes对
arXiv: K-Theory and Homology Pub Date : 2014-03-24 DOI: 10.4171/JNCG/11-3-2
Safdar Quddus
{"title":"Cohomology of $mathcal A_theta^{alg} rtimes mathbb Z_2$ and its Chern-Connes pairing","authors":"Safdar Quddus","doi":"10.4171/JNCG/11-3-2","DOIUrl":"https://doi.org/10.4171/JNCG/11-3-2","url":null,"abstract":"We calculate the Hochschild and cyclic cohomology of the $mathbb Z_2$ toroidal orbifold $mathcal A_theta^{alg} rtimes mathbb Z_2$. We also calculate the Chern-Connes pairing of the even cyclic cohomology group with the known elements of $K_0(mathcal A_theta^{alg} rtimes mathbb Z_2)$.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125889909","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
Finite generation and continuity of topological Hochschild and cyclic homology 拓扑Hochschild与循环同调的有限生成与连续性
arXiv: K-Theory and Homology Pub Date : 2014-03-03 DOI: 10.24033/ASENS.2319
B. Dundas, M. Morrow
{"title":"Finite generation and continuity of topological Hochschild and cyclic homology","authors":"B. Dundas, M. Morrow","doi":"10.24033/ASENS.2319","DOIUrl":"https://doi.org/10.24033/ASENS.2319","url":null,"abstract":"The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $TR^r$, are finitely generated modules. We use this to establish the continuity of these homology theories for any given ideal. A consequence of such continuity results is the pro Hochschild-Kostant-Rosenberg theorem for topological Hochschild and cyclic homology. Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124013797","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: 10
Index theory for manifolds with Baas-Sullivan singularities 具有Baas-Sullivan奇点流形的指标理论
arXiv: K-Theory and Homology Pub Date : 2014-02-20 DOI: 10.4171/JNCG/269
R. Deeley
{"title":"Index theory for manifolds with Baas-Sullivan singularities","authors":"R. Deeley","doi":"10.4171/JNCG/269","DOIUrl":"https://doi.org/10.4171/JNCG/269","url":null,"abstract":"We study index theory for manifolds with Baas-Sullivan singularities using geometric K-homology with coefficients in a unital C*-algebra. In particular, we define a natural analog of the Baum-Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on k-points (i.e., z/k-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed-Melrose index theorem; in the case of latter, the index theorem is related to work of Rosenberg.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114077601","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
Tate Objects in Exact Categories (with appendix by Jan vSvtov'ıvcek and Jan Trlifaj) 精确类别中的泰特对象(附Jan vSvtov vcek和Jan Trlifaj的附录)
arXiv: K-Theory and Homology Pub Date : 2014-02-20 DOI: 10.17323/1609-4514-2016-16-3-433-504
O. Braunling, M. Groechenig, J. Wolfson
{"title":"Tate Objects in Exact Categories (with appendix by Jan vSvtov'ıvcek and Jan Trlifaj)","authors":"O. Braunling, M. Groechenig, J. Wolfson","doi":"10.17323/1609-4514-2016-16-3-433-504","DOIUrl":"https://doi.org/10.17323/1609-4514-2016-16-3-433-504","url":null,"abstract":"We study elementary Tate objects in an exact category. We characterize the category of elementary Tate objects as the smallest sub-category of admissible Ind-Pro objects which contains the categories of admissible Ind-objects and admissible Pro-objects, and which is closed under extensions. We compare Beilinson's approach to Tate modules to Drinfeld's. We establish several properties of the Sato Grassmannian of an elementary Tate object in an idempotent complete exact category (e.g. it is a directed poset). We conclude with a brief treatment of n-Tate modules and n-dimensional ad`{e}les. \u0000An appendix due to J. vSvtov'ivcek and J. Trlifaj identifies the category of flat Mittag-Leffler modules with the idempotent completion of the category of admissible Ind-objects in the category of finitely generated projective modules.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121587438","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: 23
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