{"title":"TR 的色度消失结果","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16840\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16840","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
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]。
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 Lnp,fL_n^{p,f}-acyclic E1\mathbb {E}_1-rings for every 1≤k≤n1 \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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