We study relation between left and right adjoint functors to the precomposition functor. As a cosnequence we obtain various dualities in the Ext-groups in the category of strict polynomial functors.
{"title":"Poincar'e duality for Ext-groups between strict polynomial fucnctors","authors":"Marcin Chałupnik","doi":"10.1090/PROC12782","DOIUrl":"https://doi.org/10.1090/PROC12782","url":null,"abstract":"We study relation between left and right adjoint functors to the precomposition functor. As a cosnequence we obtain various dualities in the Ext-groups in the category of strict polynomial functors.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129459157","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 self-contained and purely combinatorial proof of the well known fact that the cohomology of the braces operad is the operad $mathsf{Ger}$ governing Gerstenhaber algebras.
{"title":"A Direct Computation of the Cohomology of the Braces Operad","authors":"V. Dolgushev, T. Willwacher","doi":"10.1515/FORUM-2016-0123","DOIUrl":"https://doi.org/10.1515/FORUM-2016-0123","url":null,"abstract":"We give a self-contained and purely combinatorial proof of the well known fact that the cohomology of the braces operad is the operad $mathsf{Ger}$ governing Gerstenhaber algebras.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130047745","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 introduce a Grothendieck group $E_n$ for bounded polytopes in $mathbb R^n$. It differs from the usual Euclidean scissors congruence group in that lower-dimensional polytopes are not ignored. We also define an analogous group $L_n$ using germs of polytopes at a point, which is related to spherical scissors congruence. This provides a setting for a generalization of the Dehn invariant.
{"title":"Scissors Congruence with Mixed Dimensions","authors":"T. Goodwillie","doi":"10.1090/conm/682/13806","DOIUrl":"https://doi.org/10.1090/conm/682/13806","url":null,"abstract":"We introduce a Grothendieck group $E_n$ for bounded polytopes in $mathbb R^n$. It differs from the usual Euclidean scissors congruence group in that lower-dimensional polytopes are not ignored. We also define an analogous group $L_n$ using germs of polytopes at a point, which is related to spherical scissors congruence. This provides a setting for a generalization of the Dehn invariant.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132125486","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 introduce the notion of the joint spectral flow, which is a generalization of the spectral flow, by using Segal's model of the connective $K$-theory spectrum. We apply it for some localization results of indices motivated by Witten's deformation of Dirac operators and rephrase some analytic techniques in terms of topology.
{"title":"The Joint Spectral Flow and Localization of the Indices of Elliptic Operators","authors":"Yosuke Kubota","doi":"10.2140/akt.2016.1.43","DOIUrl":"https://doi.org/10.2140/akt.2016.1.43","url":null,"abstract":"We introduce the notion of the joint spectral flow, which is a generalization of the spectral flow, by using Segal's model of the connective $K$-theory spectrum. We apply it for some localization results of indices motivated by Witten's deformation of Dirac operators and rephrase some analytic techniques in terms of topology.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132239805","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 associate to a full flag $mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $mu_{mathcal{F}}:K(F_X) to Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(cdot)$ is the $K$-theory spectrum. We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is $0$. Examining this result on the level of $K$-groups, we re-obtain various "reciprocity laws". Namely, when $X$ is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carr`{e}re reciprocity. When $X$ is higher-dimensional, we obtain Parshin reciprocity.
我们将一个“符号映射”$mu_{mathcal{F}}:K(F_X) to Sigma^n K(k)$关联到字段$k$上的一个$n$维变量$X$中的一个完整标志$mathcal{F}$。这里，$F_X$是$X$上的有理函数场，$K(cdot)$是$K$ -理论谱。我们证明了这些符号的“互惠定律”:给定一个部分标志，它的所有完整标志的和等于$0$。在$K$ -组的层次上检验这一结果，我们重新得到了各种“互易律”。即当$X$为光滑完全曲线时，得到主因子的阶为零、Weil互易性、残数定理、contou - carr互易性。当$X$是高维时，我们得到Parshin互易性。
{"title":"Reciprocity laws and K-theory","authors":"Evgeny Musicantov, Alexander Yom Din","doi":"10.2140/akt.2017.2.27","DOIUrl":"https://doi.org/10.2140/akt.2017.2.27","url":null,"abstract":"We associate to a full flag $mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a \"symbol map\" $mu_{mathcal{F}}:K(F_X) to Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(cdot)$ is the $K$-theory spectrum. We prove a \"reciprocity law\" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is $0$. Examining this result on the level of $K$-groups, we re-obtain various \"reciprocity laws\". Namely, when $X$ is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carr`{e}re reciprocity. When $X$ is higher-dimensional, we obtain Parshin reciprocity.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134580245","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 paper works out in detail the closed multicategory structure of the category of Waldhausen categories.
本文详细研究了瓦尔德豪森范畴的闭多范畴结构。
{"title":"The category of Waldhausen categories as a closed multicategory","authors":"I. Zakharevich","doi":"10.1090/CONM/707/14259","DOIUrl":"https://doi.org/10.1090/CONM/707/14259","url":null,"abstract":"This paper works out in detail the closed multicategory structure of the category of Waldhausen categories.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114472075","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 K-FJCw holds for certain subgroups of Aut($F_n$) constructed from Hol($F_2$).
我们证明了K-FJCw对于由Hol($F_2$)构造的Aut($F_n$)的某些子群成立。
{"title":"On the K-theory of certain extensions of free groups","authors":"V. Metaftsis, S. Prassidis","doi":"10.1515/FORUM-2014-0214","DOIUrl":"https://doi.org/10.1515/FORUM-2014-0214","url":null,"abstract":"We show that the K-FJCw holds for certain subgroups of Aut($F_n$) constructed from Hol($F_2$).","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129429869","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 : 2014-06-25DOI: 10.4310/HHA.2015.V17.N1.A17
Enxin Wu
Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M.~Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in~cite{I1,I2}. K.~Costello and O.~Gwilliam developed homological algebra for differentiable diffeological vector spaces in Appendix A of their book~cite{CG}. In this paper, we present homological algebra of general diffeological vector spaces via the projective objects with respect to all linear subductions, together with some applications in analysis.
{"title":"Homological Algebra for Diffeological Vector Spaces","authors":"Enxin Wu","doi":"10.4310/HHA.2015.V17.N1.A17","DOIUrl":"https://doi.org/10.4310/HHA.2015.V17.N1.A17","url":null,"abstract":"Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M.~Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in~cite{I1,I2}. K.~Costello and O.~Gwilliam developed homological algebra for differentiable diffeological vector spaces in Appendix A of their book~cite{CG}. In this paper, we present homological algebra of general diffeological vector spaces via the projective objects with respect to all linear subductions, together with some applications in analysis.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123580963","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}
For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of various generalized cohomology groups (like K-theory) and improves on terms of a spectral sequence and Gersten complexes.
{"title":"Finite homological dimension and a derived equivalence","authors":"William T. Sanders, Sarang Sane","doi":"10.1090/TRAN/6882","DOIUrl":"https://doi.org/10.1090/TRAN/6882","url":null,"abstract":"For a Cohen-Macaulay ring $R$, we exhibit the equivalence of the bounded derived categories of certain resolving subcategories, which, amongst other results, yields an equivalence of the bounded derived category of finite length and finite projective dimension modules with the bounded derived category of projective modules with finite length homologies. This yields isomorphisms of various generalized cohomology groups (like K-theory) and improves on terms of a spectral sequence and Gersten complexes.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131901648","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 : 2014-05-24DOI: 10.1142/S179352531750011X
Vito Felice Zenobi
In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.
{"title":"Mapping the surgery exact sequence for topological manifolds to analysis","authors":"Vito Felice Zenobi","doi":"10.1142/S179352531750011X","DOIUrl":"https://doi.org/10.1142/S179352531750011X","url":null,"abstract":"In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. \u0000This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. \u0000We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the \u0000APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. \u0000Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131448690","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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