TR 的色度消失结果

IF 0.8 3区 数学 Q2 MATHEMATICS
Liam Keenan, Jonas McCandless
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More precisely, we prove that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis k right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local TR vanishes on connective <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript n Superscript p comma f\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">L_n^{p,f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-acyclic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper E 1\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"double-struck\">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathbb {E}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rings for every <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 less-than-or-equal-to k less-than-or-equal-to n\"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">1 \\leq k \\leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and deduce consequences for connective Morava K-theory and the Thom spectra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"y left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">y(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\"> <mml:semantics> <mml:mi mathvariant=\"normal\">∞</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories which was recently established by Córdova Fedeli [<italic>Topological Hochschild homology of adic rings</italic>, Ph.D. thesis, University of Copenhagen, 2023].</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"54 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A chromatic vanishing result for TR\",\"authors\":\"Liam Keenan, Jonas McCandless\",\"doi\":\"10.1090/proc/16840\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper T left-parenthesis k right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">T(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local TR vanishes on connective <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript n Superscript p comma f\\\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">L_n^{p,f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-acyclic <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper E 1\\\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {E}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rings for every <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"1 less-than-or-equal-to k less-than-or-equal-to n\\\"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">1 \\\\leq k \\\\leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and deduce consequences for connective Morava K-theory and the Thom spectra <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"y left-parenthesis n right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">y(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal infinity\\\"> <mml:semantics> <mml:mi mathvariant=\\\"normal\\\">∞</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categories which was recently established by Córdova Fedeli [<italic>Topological Hochschild homology of adic rings</italic>, Ph.D. thesis, University of Copenhagen, 2023].</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"54 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16840\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16840","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

在本注释中,我们建立了望远镜局部拓扑限制同调 TR 的消失结果。更准确地说,我们证明了 T ( k ) T(k) 局部 TR 在每 1 ≤ k ≤ n 1 \leq k \leq n 的连通 L n p , f L_n^{p,f} -acyclic E 1 \mathbb {E}_1 -rings 上消失,并推导出连通莫拉瓦 K 理论和托姆谱 y ( n ) y(n) 的后果。证明依赖于 TR 与 K 理论上的曲线谱之间的关系,以及代数 K 理论保留了加性 ∞ \infty - 类别的无限乘积这一事实,这一事实最近由科尔多瓦-费德利 (Córdova Fedeli) 建立[adic rings 的拓扑霍赫希尔德同源性,哥本哈根大学博士论文,2023 年]。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A chromatic vanishing result for TR

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T ( k ) T(k) -local TR vanishes on connective L n p , f L_n^{p,f} -acyclic E 1 \mathbb {E}_1 -rings for every 1 k n 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y ( n ) y(n) . The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive \infty -categories which was recently established by Córdova Fedeli [Topological Hochschild homology of adic rings, Ph.D. thesis, University of Copenhagen, 2023].

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1.70
自引率
10.00%
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207
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期刊介绍: All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics. To be published in the Proceedings, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Longer papers may be submitted to the Transactions of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.
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