{"title":"初等超群方案的秩变异和𝜋-points","authors":"D. Benson, S. Iyengar, H. Krause, J. Pevtsova","doi":"10.1090/btran/74","DOIUrl":null,"url":null,"abstract":"<p>We develop a support theory for elementary supergroup schemes, over a field of positive characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than-or-slanted-equals 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>p</mml:mi>\n <mml:mo>⩾<!-- ⩾ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">p\\geqslant 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, starting with a definition of a <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi\">\n <mml:semantics>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\pi</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi\">\n <mml:semantics>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\pi</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k left-bracket t comma tau right-bracket slash left-parenthesis t Superscript p Baseline minus tau squared right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>k</mml:mi>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>τ<!-- τ --></mml:mi>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>t</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msup>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:msup>\n <mml:mi>τ<!-- τ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">k[t,\\tau ]/(t^p-\\tau ^2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\n <mml:semantics>\n <mml:mi>t</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has even degree and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"tau\">\n <mml:semantics>\n <mml:mi>τ<!-- τ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\tau</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Rank varieties and 𝜋-points for elementary supergroup schemes\",\"authors\":\"D. Benson, S. Iyengar, H. Krause, J. Pevtsova\",\"doi\":\"10.1090/btran/74\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We develop a support theory for elementary supergroup schemes, over a field of positive characteristic <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p greater-than-or-slanted-equals 3\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>p</mml:mi>\\n <mml:mo>⩾<!-- ⩾ --></mml:mo>\\n <mml:mn>3</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p\\\\geqslant 3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, starting with a definition of a <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi\\\">\\n <mml:semantics>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi\\\">\\n <mml:semantics>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"k left-bracket t comma tau right-bracket slash left-parenthesis t Superscript p Baseline minus tau squared right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>k</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>τ<!-- τ --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>t</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:msup>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:msup>\\n <mml:mi>τ<!-- τ --></mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">k[t,\\\\tau ]/(t^p-\\\\tau ^2)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"t\\\">\\n <mml:semantics>\\n <mml:mi>t</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">t</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> has even degree and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"tau\\\">\\n <mml:semantics>\\n <mml:mi>τ<!-- τ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tau</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.</p>\",\"PeriodicalId\":377306,\"journal\":{\"name\":\"Transactions of the American Mathematical Society, Series B\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-08-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/btran/74\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/74","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Rank varieties and 𝜋-points for elementary supergroup schemes
We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p⩾3p\geqslant 3, starting with a definition of a π\pi-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π\pi-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k[t,τ]/(tp−τ2)k[t,\tau ]/(t^p-\tau ^2), where tt has even degree and τ\tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.
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