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The derived Brauer map via twisted sheaves 通过扭曲滑轮导出的Brauer映射
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-09-22 DOI: 10.1007/s40062-023-00329-y
Guglielmo Nocera, Michele Pernice
{"title":"The derived Brauer map via twisted sheaves","authors":"Guglielmo Nocera,&nbsp;Michele Pernice","doi":"10.1007/s40062-023-00329-y","DOIUrl":"10.1007/s40062-023-00329-y","url":null,"abstract":"<div><p>Let <i>X</i> be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of Toën forms a group, which contains the classical Brauer group of <i>X</i> and which we call <span>(textsf{Br}^dagger (X))</span> following Lurie. Toën introduced a map <span>(phi :textsf{Br}^dagger (X)rightarrow H ^2_{acute{e }t }(X,{mathbb {G}}_{textrm{m}}))</span> which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of <span>(phi )</span> to a subgroup <span>(textsf{Br}(X)subset textsf{Br}^dagger (X))</span>, which we call the <i>derived Brauer group</i>, on which <span>(phi )</span> becomes an isomorphism <span>(textsf{Br}(X)simeq H ^2_{acute{e }t }(X,{mathbb {G}}_{textrm{m}}))</span>. This map may be interpreted as a derived version of the classical Brauer map which offers a way to “fill the gap” between the classical Brauer group and the cohomogical Brauer group. The group <span>(textsf{Br}(X))</span> was introduced by Lurie by making use of the theory of prestable <span>(infty )</span>-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of <span>(infty )</span>-categories between the <i>Brauer space</i> of invertible presentable prestable <span>({{mathcal {O}}}_X)</span>-linear categories, and the space <span>(Map (X,K ({mathbb {G}}_{textrm{m}},2)))</span>. We offer an alternative proof of this equivalence of <span>(infty )</span>-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"369 - 396"},"PeriodicalIF":0.5,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00329-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Eilenberg–Maclane spaces and stabilisation in homotopy type theory Eilenberg–Maclane空间与同构型理论中的稳定
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-09-21 DOI: 10.1007/s40062-023-00330-5
David Wärn
{"title":"Eilenberg–Maclane spaces and stabilisation in homotopy type theory","authors":"David Wärn","doi":"10.1007/s40062-023-00330-5","DOIUrl":"10.1007/s40062-023-00330-5","url":null,"abstract":"<div><p>In this note, we study the delooping of spaces and maps in homotopy type theory. We show that in some cases, spaces have a unique delooping, and give a simple description of the delooping in these cases. We explain why some maps, such as group homomorphisms, have a unique delooping. We discuss some applications to Eilenberg–MacLane spaces and cohomology.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"357 - 368"},"PeriodicalIF":0.5,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00330-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1 透镜空间上极Morse–Bott叶理微分同胚群的同胚型，1
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-08-08 DOI: 10.1007/s40062-023-00328-z
Oleksandra Khokhliuk, Sergiy Maksymenko
{"title":"Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1","authors":"Oleksandra Khokhliuk,&nbsp;Sergiy Maksymenko","doi":"10.1007/s40062-023-00328-z","DOIUrl":"10.1007/s40062-023-00328-z","url":null,"abstract":"<div><p>Let <span>(T= S^1times D^2)</span> be the solid torus, <span>(mathcal {F})</span> the Morse–Bott foliation on <i>T</i> into 2-tori parallel to the boundary and one singular circle <span>(S^1times 0)</span>, which is the central circle of the torus <i>T</i>, and <span>(mathcal {D}(mathcal {F},partial T))</span> the group of diffeomorphisms of <i>T</i> fixed on <span>(partial T)</span> and leaving each leaf of the foliation <span>(mathcal {F})</span> invariant. We prove that <span>(mathcal {D}(mathcal {F},partial T))</span> is contractible. Gluing two copies of <i>T</i> by some diffeomorphism between their boundaries, we will get a lens space <span>(L_{p,q})</span> with a Morse–Bott foliation <span>(mathcal {F}_{p,q})</span> obtained from <span>(mathcal {F})</span> on each copy of <i>T</i>. We also compute the homotopy type of the group <span>(mathcal {D}(mathcal {F}_{p,q}))</span> of diffeomorphisms of <span>(L_{p,q})</span> leaving invariant each leaf of <span>(mathcal {F}_{p,q})</span>.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"313 - 356"},"PeriodicalIF":0.5,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00328-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Goodwillie’s cosimplicial model for the space of long knots and its applications 长结空间的Goodwillie协单纯形模型及其应用
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-08-02 DOI: 10.1007/s40062-023-00327-0
Yuqing Shi
{"title":"Goodwillie’s cosimplicial model for the space of long knots and its applications","authors":"Yuqing Shi","doi":"10.1007/s40062-023-00327-0","DOIUrl":"10.1007/s40062-023-00327-0","url":null,"abstract":"<div><p>We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield–Kan homotopy spectral sequence of the tower of fibrations, given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in Conant (Am J Math 130(2):341–357. https://doi.org/10.1353/ajm.2008.0020, 2008), we give a combinatorial interpretation of the differentials <span>(d^1)</span> mapping into the diagonal terms, by introducing the notion of (<i>i</i>, <i>n</i>)-marked unitrivalent graphs.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"265 - 312"},"PeriodicalIF":0.5,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00327-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Centralisers, complex reflection groups and actions in the Weyl group (E_6) Weyl群中的中心子、复反射群和作用
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-06-08 DOI: 10.1007/s40062-023-00326-1
Graham A. Niblo, Roger Plymen, Nick Wright
{"title":"Centralisers, complex reflection groups and actions in the Weyl group (E_6)","authors":"Graham A. Niblo,&nbsp;Roger Plymen,&nbsp;Nick Wright","doi":"10.1007/s40062-023-00326-1","DOIUrl":"10.1007/s40062-023-00326-1","url":null,"abstract":"<div><p>The compact, connected Lie group <span>(E_6)</span> admits two forms: simply connected and adjoint type. As we previously established, the Baum–Connes isomorphism relates the two Langlands dual forms, giving a duality between the equivariant K-theory of the Weyl group acting on the corresponding maximal tori. Our study of the <span>(A_n)</span> case showed that this duality persists at the level of homotopy, not just homology. In this paper we compute the extended quotients of maximal tori for the two forms of <span>(E_6)</span>, showing that the homotopy equivalences of sectors established in the <span>(A_n)</span> case also exist here, leading to a conjecture that the homotopy equivalences always exist for Langlands dual pairs. In computing these sectors we show that centralisers in the <span>(E_6)</span> Weyl group decompose as direct products of reflection groups, generalising Springer’s results for regular elements, and we develop a pairing between the component groups of fixed sets generalising Reeder’s results. As a further application we compute the <i>K</i>-theory of the reduced Iwahori-spherical <span>(C^*)</span>-algebra of the p-adic group <span>(E_6)</span>, which may be of adjoint type or simply connected.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"219 - 264"},"PeriodicalIF":0.5,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00326-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A t-structure on the (infty )-category of mixed graded modules 混合分次模（infty）-范畴上的一个t-结构
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-04-27 DOI: 10.1007/s40062-023-00324-3
Emanuele Pavia
{"title":"A t-structure on the (infty )-category of mixed graded modules","authors":"Emanuele Pavia","doi":"10.1007/s40062-023-00324-3","DOIUrl":"10.1007/s40062-023-00324-3","url":null,"abstract":"<div><p>In this work, we shall study in a purely model-independent fashion the <span>(infty )</span>-category of mixed graded modules over a ring of characteristic 0, as defined by D. Calaque, T. Pantev, M. Vaquié, B. Toën and G. Vezzosi. First, we collect some basic results about its main formal properties, clarifying foundational questions in a systematic manner, to serve as a reference for future work. Finally, we shall endow such <span>(infty )</span>-category with a both left and right complete accessible <i>t</i>-structure, showing how this identifies the <span>(infty )</span>-category of mixed graded modules with the left completion of the Beilinson <i>t</i>-structure on the <span>(infty )</span>-category of filtered modules.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"177 - 218"},"PeriodicalIF":0.5,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00324-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On Kähler differentials of divided power algebras 关于分幂代数的Kähler微分
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-04-10 DOI: 10.1007/s40062-023-00325-2
Ioannis Dokas
{"title":"On Kähler differentials of divided power algebras","authors":"Ioannis Dokas","doi":"10.1007/s40062-023-00325-2","DOIUrl":"10.1007/s40062-023-00325-2","url":null,"abstract":"<div><p>The Quillen–Barr–Beck cohomology of augmented algebras with a system of divided powers is defined as the derived functor of Beck derivations. The main theorem of this paper states that the Kähler differentials of an augmented algebra with a system of divided powers in prime characteristic represents Beck derivations. We give a geometrical interpretation of this statement for the sheaf of relative differentials. As an application in the theory of modular Lie algebras we prove that any special derivation of a divided power algebra is a Beck derivation and we apply the theorem to Witt algebras.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 2-3","pages":"153 - 176"},"PeriodicalIF":0.5,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-023-00325-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41081239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Coherent presentations of monoids with a right-noetherian Garside family 具有右右Garside族的一元群的连贯表示
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-02-20 DOI: 10.1007/s40062-023-00323-4
Pierre-Louis Curien, Alen Ɖurić, Yves Guiraud
{"title":"Coherent presentations of monoids with a right-noetherian Garside family","authors":"Pierre-Louis Curien,&nbsp;Alen Ɖurić,&nbsp;Yves Guiraud","doi":"10.1007/s40062-023-00323-4","DOIUrl":"10.1007/s40062-023-00323-4","url":null,"abstract":"<div><p>This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for Artin-Tits monoids of spherical type: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 1","pages":"115 - 152"},"PeriodicalIF":0.5,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4787115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The cohomology of (C_2)-surfaces with ({underline{{mathbb {Z}}}})-coefficients (C_2) -曲面与({underline{{mathbb {Z}}}}) -系数的上同调
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-01-20 DOI: 10.1007/s40062-022-00321-y
Christy Hazel
{"title":"The cohomology of (C_2)-surfaces with ({underline{{mathbb {Z}}}})-coefficients","authors":"Christy Hazel","doi":"10.1007/s40062-022-00321-y","DOIUrl":"10.1007/s40062-022-00321-y","url":null,"abstract":"<div><p>Let <span>(C_2)</span> denote the cyclic group of order 2. We compute the <span>(RO(C_2))</span>-graded cohomology of all <span>(C_2)</span>-surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether or not the action preserves the orientation, and one other invariant.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 1","pages":"71 - 114"},"PeriodicalIF":0.5,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4788434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Modules and representations up to homotopy of Lie n-algebroids 李n -代数群的模与表示直至同伦
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-01-05 DOI: 10.1007/s40062-022-00322-x
M. Jotz, R. A. Mehta, T. Papantonis
{"title":"Modules and representations up to homotopy of Lie n-algebroids","authors":"M. Jotz,&nbsp;R. A. Mehta,&nbsp;T. Papantonis","doi":"10.1007/s40062-022-00322-x","DOIUrl":"10.1007/s40062-022-00322-x","url":null,"abstract":"<div><p>This paper studies differential graded modules and representations up to homotopy of Lie <i>n</i>-algebroids, for general <span>(nin {mathbb {N}})</span>. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie <i>n</i>-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie <i>n</i>-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie <i>n</i>-algebroids are used to encode decomposed VB-Lie <i>n</i>-algebroid structures on double vector bundles.</p></div>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 1","pages":"23 - 70"},"PeriodicalIF":0.5,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-022-00322-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4210810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
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