Journal of Pure and Applied Algebra最新文献

{"title":"Retraction notice to “On the fifth Singer algebraic transfer in a generic family of internal degree characterized by μ(n)=4” [J. Pure Appl. Algebra 228 (2024) 107658]","authors":"Đặng Võ Phúc","doi":"10.1016/j.jpaa.2025.107873","DOIUrl":"10.1016/j.jpaa.2025.107873","url":null,"abstract":"<div><div>This article has been retracted: please see Elsevier Policy on Article Withdrawal (<span><span>https://www.elsevier.com/about/policies/article-withdrawal</span><svg><path></path></svg></span>).</div><div>This article has been retracted at the request of the Editors-in-Chief.</div><div>Many of the results in this paper are false, including the main result (Theorem 1.3), whose proof has a mistake, and several of its consequences are asserted without proof. In a letter to the Editors providing comments to the paper, a colleague pointed out that a proof is only provided for the cases <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>8</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, asserting that the main result is false for <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>8</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and also incomplete for <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. No proof is given for <span><math><mi>s</mi><mo>&gt;</mo><mn>2</mn></math></span>.</div><div>A detailed assessment by independent referees created reasonable doubt that the author's results were correct.</div><div>The Corrigendum to this article does not provide any proofs, but points out an “inaccuracy” in a different part of the paper (Cor. 1.11).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107873"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Directed graphs, Frattini-resistance, and maximal pro-p Galois groups","authors":"Claudio Quadrelli","doi":"10.1016/j.jpaa.2024.107857","DOIUrl":"10.1016/j.jpaa.2024.107857","url":null,"abstract":"<div><div>Let <em>p</em> be a prime. Following Snopce-Tanushevski, a pro-<em>p</em> group <em>G</em> is called Frattini-resistant if the function <span><math><mi>H</mi><mo>↦</mo><mi>Φ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, from the poset of all closed topologically finitely generated subgroups of <em>G</em> into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro-<em>p</em> group (oriented pro-<em>p</em> RAAG) <em>G</em> associated to a finite directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; <em>G</em> is Frattini-resistant; every topologically finitely generated closed subgroup of <em>G</em> is an oriented pro-<em>p</em> RAAG; <em>G</em> is the maximal pro-<em>p</em> Galois group of a field containing a root of 1 of order <em>p</em>. Also, we conjecture that in the <span><math><mi>Z</mi><mo>/</mo><mi>p</mi></math></span>-cohomology of a Frattini-resistant pro-<em>p</em> group there are no essential triple Massey products.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107857"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The multiple holomorph of centerless groups","authors":"Cindy (Sin Yi) Tsang","doi":"10.1016/j.jpaa.2024.107843","DOIUrl":"10.1016/j.jpaa.2024.107843","url":null,"abstract":"<div><div>Let <em>G</em> be a group. The holomorph <span><math><mrow><mi>Hol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of <em>G</em>. The multiple holomorph <span><math><mrow><mi>NHol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is in turn defined as the normalizer of the holomorph. Their quotient <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mrow><mi>NHol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mrow><mi>Hol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has been computed for various families of groups <em>G</em>. In this paper, we consider the case when <em>G</em> is centerless, and we show that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> must have exponent at most 2 unless <em>G</em> satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has order 2 for all almost simple groups <em>G</em>, and that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has exponent at most 2 for all centerless perfect or complete groups <em>G</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107843"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relative extensions and cohomology of profinite groups","authors":"Gareth Wilkes","doi":"10.1016/j.jpaa.2024.107846","DOIUrl":"10.1016/j.jpaa.2024.107846","url":null,"abstract":"<div><div>We construct a correspondence between the cohomology groups of a group <em>G</em> relative to a family of subgroups <span><math><mi>S</mi></math></span> and the classes of ‘relative extensions’ of <em>G</em> by abelian groups, modulo a certain equivalence relation. We establish this correspondence for both discrete and profinite group pairs. We go on to discuss the relationships of profinite group pairs of cohomological dimension one with free products and projective group pairs.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107846"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On irreducibility of modules of Whittaker type: Twisted modules and nonabelian orbifolds","authors":"Dražen Adamović ,&nbsp;Ching Hung Lam ,&nbsp;Veronika Pedić Tomić ,&nbsp;Nina Yu","doi":"10.1016/j.jpaa.2024.107840","DOIUrl":"10.1016/j.jpaa.2024.107840","url":null,"abstract":"<div><div>In <span><span>[1]</span></span>, we extended the Dong-Mason theorem on irreducibility of modules for cyclic orbifold vertex algebras (cf. <span><span>[12]</span></span>) to the entire category of weak modules and applied this result to Whittaker modules. In this paper, we present further generalizations of these results for nonabelian orbifolds of vertex operator superalgebras. Let <em>V</em> be a vertex superalgebra of a countable dimension and let <em>G</em> be a finite subgroup of <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>V</mi><mo>)</mo></math></span>. Assume that <span><math><mi>h</mi><mo>∈</mo><mi>Z</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> where <span><math><mi>Z</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the center of the group <em>G</em>. For any irreducible <em>h</em>–twisted (weak) <em>V</em>–module <em>M</em>, we prove that if <span><math><mi>M</mi><mo>≇</mo><mi>g</mi><mo>∘</mo><mi>M</mi></math></span> for all <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span> then <em>M</em> is also irreducible as <span><math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span>–module. We also apply this result to examples and give irreducibility of modules of Whittaker type for orbifolds of Neveu-Schwarz vertex superalgebras, Heisenberg vertex algebras, Virasoro vertex operator algebra and Heisenberg-Virasoro vertex algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107840"},"PeriodicalIF":0.7,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the cohomology of Lie algebras associated with graphs","authors":"Marco Aldi ,&nbsp;Andrew Butler ,&nbsp;Jordan Gardiner ,&nbsp;Daniele Grandini ,&nbsp;Monica Lichtenwalner ,&nbsp;Kevin Pan","doi":"10.1016/j.jpaa.2024.107838","DOIUrl":"10.1016/j.jpaa.2024.107838","url":null,"abstract":"<div><div>We describe a canonical decomposition of the cohomology of the Dani-Mainkar 2-step nilpotent Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the cohomology in all degrees of Lie algebras associated with arbitrary star graphs. We also describe a procedure to reduce the calculation of the cohomology of solvable Lie algebras associated with graphs through the Grantcharov-Grantcharov-Iliev construction to the cohomology of Dani-Mainkar Lie algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107838"},"PeriodicalIF":0.7,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Normalizer quotients of symmetric groups and inner holomorphs","authors":"Alexei Entin ,&nbsp;Cindy (Sin Yi) Tsang","doi":"10.1016/j.jpaa.2024.107839","DOIUrl":"10.1016/j.jpaa.2024.107839","url":null,"abstract":"<div><div>We show that every finite group <em>T</em> is isomorphic to a normalizer quotient <span><math><msub><mrow><mi>N</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>/</mo><mi>H</mi></math></span> for some <em>n</em> and a subgroup <span><math><mi>H</mi><mo>≤</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We show that this holds for all large enough <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> and also with <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> replaced by <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group <em>G</em> with <span><math><mrow><mi>Out</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>≅</mo><mi>T</mi></math></span> (for any given finite group <em>T</em>) and the determination of the normalizer in <span><math><mrow><mi>Sym</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of the inner holomorph <span><math><mrow><mi>InHol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mrow><mi>Sym</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for any centerless indecomposable finite group <em>G</em>, which may be of independent interest.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107839"},"PeriodicalIF":0.7,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Laumon parahoric local models via quiver Grassmannians","authors":"Evgeny Feigin ,&nbsp;Martina Lanini ,&nbsp;Alexander Pütz","doi":"10.1016/j.jpaa.2024.107837","DOIUrl":"10.1016/j.jpaa.2024.107837","url":null,"abstract":"<div><div>Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107837"},"PeriodicalIF":0.7,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Period integrals of smooth projective complete intersections as exponential periods","authors":"Jeehoon Park","doi":"10.1016/j.jpaa.2024.107836","DOIUrl":"10.1016/j.jpaa.2024.107836","url":null,"abstract":"<div><div>Let <em>X</em> be a smooth projective complete intersection over <span><math><mi>Q</mi></math></span> of dimension <span><math><mi>n</mi><mo>−</mo><mi>k</mi></math></span> in the projective space <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> defined by the zero locus of <span><math><munder><mrow><mi>f</mi></mrow><mo>_</mo></munder><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo><mo>)</mo></math></span>, for given positive integers <em>n</em> and <em>k</em>. For a given primitive homology cycle <span><math><mo>[</mo><mi>γ</mi><mo>]</mo><mo>∈</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mrow><mo>(</mo><mi>X</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>,</mo><mi>Z</mi><mo>)</mo></mrow><mrow><mn>0</mn></mrow></msub></math></span>, the period integral is defined to be a linear map from the primitive de Rham cohomology group <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>d</mi><mi>R</mi><mo>,</mo><mi>prim</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>;</mo><mi>Q</mi><mo>)</mo></math></span> to <span><math><mi>C</mi></math></span> given by <span><math><mo>[</mo><mi>ω</mi><mo>]</mo><mo>↦</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>γ</mi></mrow></msub><mi>ω</mi></math></span>. The goal of this article is to interpret this period integral as <em>Feynman's path integral</em> of 0-dimensional quantum field theory with the action functional <span><math><mi>S</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mrow><mi>y</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><msub><mrow><mi>f</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo></math></span> (in other words, <em>the exponential period</em> with the action functional <em>S</em>) and use this interpretation to develop a formal deformation theory of period integrals of <em>X</em>, which can be viewed as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107836"},"PeriodicalIF":0.7,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"GT-shadows for the gentle version GTˆgen of the Grothendieck-Teichmueller group","authors":"Vasily A. Dolgushev ,&nbsp;Jacob J. Guynee","doi":"10.1016/j.jpaa.2024.107819","DOIUrl":"10.1016/j.jpaa.2024.107819","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> be the Artin braid group on 3 strands and <span><math><msub><mrow><mi>PB</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> be the corresponding pure braid group. In this paper, we construct the groupoid <span><math><mi>GTSh</mi></math></span> of <span><math><mi>GT</mi></math></span>-shadows for a (possibly more tractable) version <span><math><msub><mrow><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msub></math></span> of the Grothendieck-Teichmueller group <span><math><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> introduced in paper <span><span>[12]</span></span> by D. Harbater and L. Schneps. We call this group the gentle version of <span><math><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> and denote it by <span><math><msub><mrow><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></math></span>. The objects of <span><math><mi>GTSh</mi></math></span> are finite index normal subgroups <span><math><mi>N</mi></math></span> of <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> satisfying the condition <span><math><mi>N</mi><mo>≤</mo><msub><mrow><mi>PB</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Morphisms of <span><math><mi>GTSh</mi></math></span> are called <span><math><mi>GT</mi></math></span>-shadows and they may be thought of as approximations to elements of <span><math><msub><mrow><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></math></span>. We show how <span><math><mi>GT</mi></math></span>-shadows can be obtained from elements of <span><math><msub><mrow><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></math></span> and prove that <span><math><msub><mrow><mover><mrow><mi>GT</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></math></span> is isomorphic to the limit of a certain functor defined in terms of the groupoid <span><math><mi>GTSh</mi></math></span>. Using this result, we get a criterion for identifying genuine <span><math><mi>GT</mi></math></span>-shadows.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107819"},"PeriodicalIF":0.7,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}