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{"title":"分级的精确序列","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":50188,"journal":{"name":"Journal of Group Theory","volume":"15 38","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":50188,\"journal\":{\"name\":\"Journal of Group Theory\",\"volume\":\"15 38\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-11-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Group Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/jgth-2023-0040\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0040","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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An exact sequence for the graded Picent
To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picent gr ( 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 Picent gr ( A ) \mathrm{Picent}^{\mathrm{gr}}(A) , Picent ( B ) \mathrm{Picent}(B) , and group cohomology.
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