{"title":"An exact sequence for the graded Picent","authors":"Andrei Marcus, Virgilius-Aurelian Minuță","doi":"10.1515/jgth-2023-0040","DOIUrl":null,"url":null,"abstract":"To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0001.png\" /> <jats:tex-math>\\mathrm{Picent}^{\\mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of isomorphism classes of invertible 𝐺-graded <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0002.png\" /> <jats:tex-math>(A,A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>Picent</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0003.png\" /> <jats:tex-math>\\mathrm{Picent}</jats:tex-math> </jats:alternatives> </jats:inline-formula> version of the Beattie–del Río exact sequence, involving Dade’s group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0004.png\" /> <jats:tex-math>G[B]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which relates <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0001.png\" /> <jats:tex-math>\\mathrm{Picent}^{\\mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Picent</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0006.png\" /> <jats:tex-math>\\mathrm{Picent}(B)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and group cohomology.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0040","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picentgr(A)\mathrm{Picent}^{\mathrm{gr}}(A) of isomorphism classes of invertible 𝐺-graded (A,A)(A,A)-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a Picent\mathrm{Picent} version of the Beattie–del Río exact sequence, involving Dade’s group G[B]G[B], which relates Picentgr(A)\mathrm{Picent}^{\mathrm{gr}}(A), Picent(B)\mathrm{Picent}(B), and group cohomology.
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