{"title":"𝑅𝑂(𝐶₂)-等级化的𝑅𝑂(𝐶₂)-广义稳定茎的还原环","authors":"Eva Belmont, Zhouli Xu, Shangjie Zhang","doi":"10.1090/bproc/203","DOIUrl":null,"url":null,"abstract":"<p>We describe in terms of generators and relations the ring structure of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R upper O left-parenthesis upper C 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>R</mml:mi>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">RO(C_2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-graded <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">C_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-equivariant stable stems <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript star Superscript upper C 2\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mo>⋆<!-- ⋆ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">\\pi _\\star ^{C_2}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">C_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-equivariant sphere <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript star Superscript upper C 2 Baseline left-parenthesis double-struck upper S Subscript double-struck upper Q Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mo>⋆<!-- ⋆ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msub>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n </mml:mrow>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">S</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\pi _\\star ^{C_2}(\\mathbb {S}_\\mathbb {Q})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"50 6","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems\",\"authors\":\"Eva Belmont, Zhouli Xu, Shangjie Zhang\",\"doi\":\"10.1090/bproc/203\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We describe in terms of generators and relations the ring structure of the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R upper O left-parenthesis upper C 2 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>R</mml:mi>\\n <mml:mi>O</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">RO(C_2)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-graded <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C_2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-equivariant stable stems <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi Subscript star Superscript upper C 2\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mo>⋆<!-- ⋆ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:msub>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi _\\\\star ^{C_2}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C_2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-equivariant sphere <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi Subscript star Superscript upper C 2 Baseline left-parenthesis double-struck upper S Subscript double-struck upper Q Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msubsup>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mo>⋆<!-- ⋆ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:msub>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n </mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">S</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi _\\\\star ^{C_2}(\\\\mathbb {S}_\\\\mathbb {Q})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"50 6\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/203\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/203","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们用生成器和关系来描述 R O ( C 2 ) RO(C_2) -等级的 C 2 C_2 -等价稳定茎 π ⋆ C 2 \pi _\star ^{C_2} modulo the ideal of all nilpotent elements 的环结构。因此，我们还记录了有理 C 2 C_2 -等价球 π ⋆ C 2 ( S Q ) \pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q}) 的同调群的环结构。
The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems
We describe in terms of generators and relations the ring structure of the RO(C2)RO(C_2)-graded C2C_2-equivariant stable stems π⋆C2\pi _\star ^{C_2} modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational C2C_2-equivariant sphere π⋆C2(SQ)\pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q}).
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