{"title":"高半加性Grothendieck-Witt理论与𝐾(1)局部球","authors":"Shachar Carmeli, Allen Yuan","doi":"10.1090/cams/17","DOIUrl":null,"url":null,"abstract":"<p>We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">K(1)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-local sphere <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\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>K</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {S}_{K(1)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> at the prime <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, in particular realizing the non-<inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-adic rational element <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 plus epsilon element-of pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mo>+</mml:mo>\n <mml:mi>ε<!-- ε --></mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:msub>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\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>K</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">1+\\varepsilon \\in \\pi _0\\mathbb {S}_{K(1)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\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>K</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\pi _0\\mathbb {S}_{K(1)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"248 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere\",\"authors\":\"Shachar Carmeli, Allen Yuan\",\"doi\":\"10.1090/cams/17\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>K</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">K(1)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-local sphere <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\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>K</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {S}_{K(1)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> at the prime <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, in particular realizing the non-<inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-adic rational element <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1 plus epsilon element-of pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>1</mml:mn>\\n <mml:mo>+</mml:mo>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:msub>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\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>K</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">1+\\\\varepsilon \\\\in \\\\pi _0\\\\mathbb {S}_{K(1)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>π<!-- π --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\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>K</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\pi _0\\\\mathbb {S}_{K(1)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":285678,\"journal\":{\"name\":\"Communications of the American Mathematical Society\",\"volume\":\"248 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-09-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications of the American Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/cams/17\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications of the American Mathematical 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Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere
We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the K(1)K(1)-local sphere SK(1)\mathbb {S}_{K(1)} at the prime 22, in particular realizing the non-22-adic rational element 1+ε∈π0SK(1)1+\varepsilon \in \pi _0\mathbb {S}_{K(1)} as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in π0SK(1)\pi _0\mathbb {S}_{K(1)}.
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