I present a proof of Kirchberg’s classification theorem: two separable, nuclear, O∞mathcal O_infty-stable C∗C^ast-algebras are stably isomorphic if and only if they are ideal-related KKKK-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.
我提出了基希贝格分类定理的一个证明：当且仅当两个可分离的、核的、O ∞ mathcal O_infty -stable C ∗ C^ast -gealbras 是理想相关的 K KK -equivalent 时，它们是稳定同构的。特别是，这为基希贝格-菲利普斯定理提供了一个更基本的证明，该定理在论文中被单独列出，以增加这一重要特例的可读性。
{"title":"Classification of 𝒪_{∞}-Stable 𝒞*-Algebras","authors":"James Gabe","doi":"10.1090/memo/1461","DOIUrl":"https://doi.org/10.1090/memo/1461","url":null,"abstract":"<p>I present a proof of Kirchberg’s classification theorem: two separable, nuclear, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal O_infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-stable <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^ast</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebras are stably isomorphic if and only if they are ideal-related <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K upper K\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>K</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">KK</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" 821","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139391793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the second paper in a series of papers analyzing angled crested like water waves with surface tension. We consider the 2D capillary gravity water wave equation and assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the first paper cite{Ag19} we constructed a weighted energy which generalizes the energy of Kinsey and Wu cite{KiWu18} to the case of non-zero surface tension, and proved a local wellposedness result. In this paper we prove that under a suitable scaling regime, the zero surface tension limit of these solutions with surface tension are solutions to the gravity water wave equation which includes waves with angled crests.
{"title":"Angled Crested Like Water Waves with Surface Tension II: Zero Surface Tension Limit","authors":"Siddhant Agrawal","doi":"10.1090/memo/1458","DOIUrl":"https://doi.org/10.1090/memo/1458","url":null,"abstract":"This is the second paper in a series of papers analyzing angled crested like water waves with surface tension. We consider the 2D capillary gravity water wave equation and assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the first paper cite{Ag19} we constructed a weighted energy which generalizes the energy of Kinsey and Wu cite{KiWu18} to the case of non-zero surface tension, and proved a local wellposedness result. In this paper we prove that under a suitable scaling regime, the zero surface tension limit of these solutions with surface tension are solutions to the gravity water wave equation which includes waves with angled crests.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"149 3","pages":""},"PeriodicalIF":1.9,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139395639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>We show that the generation problem in Thompson’s group <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula> is decidable, i.e., there is an algorithm which decides if a finite set of elements of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula> generates the whole <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula>. The algorithm makes use of the Stallings <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics></mml:math></inline-formula>-core of subgroups of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula>, which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics></mml:math></inline-formula>-core of subgroups of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula> provides a solution to another algorithmic problem in <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics></mml:math></inline-formula>. Namely, given a finitely generated subgroup <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics></mml:math></inline-formula> of <inline-formul
{"title":"The Generation Problem in Thompson Group 𝐹","authors":"Gili Golan Polak","doi":"10.1090/memo/1451","DOIUrl":"https://doi.org/10.1090/memo/1451","url":null,"abstract":"<p>We show that the generation problem in Thompson’s group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is decidable, i.e., there is an algorithm which decides if a finite set of elements of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> generates the whole <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The algorithm makes use of the Stallings <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-core of subgroups of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-core of subgroups of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> provides a solution to another algorithmic problem in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Namely, given a finitely generated subgroup <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formul","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" 40","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138612675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chris Parker, Gerald Pientka, Andreas Seidel, G. Stroth
<p>Let <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula> be a prime. In this paper we investigate finite <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K Subscript StartSet 2 comma p EndSet"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">K</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">mathcal K_{{2,p}}</mml:annotation> </mml:semantics></mml:math></inline-formula>-groups <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics></mml:math></inline-formula> which have a subgroup <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H less-than-or-equal-to upper G"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">H le G</mml:annotation> </mml:semantics></mml:math></inline-formula> such that <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K less-than-or-equal-to upper H equals upper N Subscript upper G Baseline left-parenthesis upper K right-parenthesis less-than-or-equal-to upper A u t left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>Aut</mml:mi> <mml:mo><!-- --></mml:mo> <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">K le H = N_G(K) le operatorname {Aut}(K)</mml:annotation> </mml:semantics></mml:math></inline-formula> for <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotatio
设p是素数。本文研究有限K {2，p} 数学K_{{2,p}} -群G G，其子群H≤G H le G使得K≤H = N G(K)≤Aut (K) K le H = N_G(K) le 算子名{Aut}(K)对于K K是特征为p p的李型的简单群，且| G:H| |G:H|是p p的素数。如果G G具有局部特征p p，则G G在特征p p上几乎是李氏型。这里G G具有局部特征p p意味着对于所有非平凡p -子群p p (G G)和Q Q (N G(p) N_G(p)中最大的正规p -子群)我们有包含C G(Q)≤Q C_G(Q)le Q。我们确定了特征p p几乎为李氏型的群的结构细节。特别地，在K K的秩至少为33的情况下，我们证明了G =
{"title":"Finite Groups Which are Almost Groups of Lie Type in Characteristic 𝐩","authors":"Chris Parker, Gerald Pientka, Andreas Seidel, G. Stroth","doi":"10.1090/memo/1452","DOIUrl":"https://doi.org/10.1090/memo/1452","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a prime. In this paper we investigate finite <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper K Subscript StartSet 2 comma p EndSet\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">K</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal K_{{2,p}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which have a subgroup <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H less-than-or-equal-to upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H le G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> such that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K less-than-or-equal-to upper H equals upper N Subscript upper G Baseline left-parenthesis upper K right-parenthesis less-than-or-equal-to upper A u t left-parenthesis upper K right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>Aut</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K le H = N_G(K) le operatorname {Aut}(K)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotatio","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" 417","pages":""},"PeriodicalIF":1.9,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138611071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript q Baseline left-parenthesis upper G Subscript m n Baseline left-parenthesis double-struck upper F right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{mathcal O}_q(G_{mn}(mathbb {F}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript q Baseline left-parenthesis upper G Subscript m n Baseline left-parenthesis double-struck upper F right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{mathcal O}_q(G_{mn}(mathbb {F}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and prove a version of the orbit method for torus-invariant objects. Finally,
{"title":"Total Positivity is a Quantum Phenomenon: The Grassmannian Case","authors":"Stéphane Launois, Tom Lenagan, Brendan Nolan","doi":"10.1090/memo/1448","DOIUrl":"https://doi.org/10.1090/memo/1448","url":null,"abstract":"The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=\"application/x-tex\">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript q Baseline left-parenthesis upper G Subscript m n Baseline left-parenthesis double-struck upper F right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">F</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathcal O}_q(G_{mn}(mathbb {F}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript q Baseline left-parenthesis upper G Subscript m n Baseline left-parenthesis double-struck upper F right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">F</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathcal O}_q(G_{mn}(mathbb {F}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and prove a version of the orbit method for torus-invariant objects. Finally, ","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"231 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define an infinite sequence of generalizations, parametrized by an integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of the Stieltjes–Rogers and Thron–Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dyck and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of polynomials. Enumeration of unlabeled ordered trees and forests gives rise to multivariate Fuss–Narayana polynomials and Fuss–Narayana symmetric functions. Enumeration of increasing (labeled) ordered trees and forests gives rise to multivariate Eulerian polynomials and Eulerian symmetric functions, which include the univariate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th-order Eulerian polynomials as specializations. We also find branched continued fractions for ratios of contiguous hypergeometric series <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript r Baseline upper F Subscript s"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mi>r</mml:mi> </mml:msub> <mml:mspace width="negativethinmathspace" /> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{}_r ! F_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for arbitrary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-t
我们定义了stieltje - rogers和thrn - rogers多项式的无限推广序列，参数化为整数m≥1 m ge 1;它们是一些分支连分式的幂级数展开式，以及m m m -Dyck和m m -Schröder具有高度相关权重路径的生成多项式。我们证明了所有这些多项式序列在所有(无穷多个)不定式中都是系数上的汉克尔完全正的。然后，我们应用该理论证明了组合感兴趣多项式序列的系数方向上的汉克尔全正性。未标记有序树和森林的枚举产生多元的Fuss-Narayana多项式和Fuss-Narayana对称函数。增加(标记)有序树木和森林的枚举产生多元欧拉多项式和欧拉对称函数，其中包括作为专门化的单变量mm阶欧拉多项式。我们还发现了连续超几何级数r F s {}＿r !F＿s对于任意r r和ss，它推广了高斯连续分数对于连续的2个F 1{} ＿2 !F＿1;当s=0时，证明了系数的汉克尔全正性。最后，我们将分支连分数推广到连续的基本超几何级数r φ s {}＿r !phi ＿s。
{"title":"Lattice Paths and Branched Continued Fractions: An Infinite Sequence of Generalizations of the Stieltjes–Rogers and Thron–Rogers Polynomials, with Coefficientwise Hankel-Total Positivity","authors":"Mathias Pétréolle, Alan D. Sokal, Bao-Xuan Zhu","doi":"10.1090/memo/1450","DOIUrl":"https://doi.org/10.1090/memo/1450","url":null,"abstract":"We define an infinite sequence of generalizations, parametrized by an integer <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m greater-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">m ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of the Stieltjes–Rogers and Thron–Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=\"application/x-tex\">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dyck and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=\"application/x-tex\">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of polynomials. Enumeration of unlabeled ordered trees and forests gives rise to multivariate Fuss–Narayana polynomials and Fuss–Narayana symmetric functions. Enumeration of increasing (labeled) ordered trees and forests gives rise to multivariate Eulerian polynomials and Eulerian symmetric functions, which include the univariate <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=\"application/x-tex\">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th-order Eulerian polynomials as specializations. We also find branched continued fractions for ratios of contiguous hypergeometric series <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"Subscript r Baseline upper F Subscript s\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> </mml:mrow> <mml:mi>r</mml:mi> </mml:msub> <mml:mspace width=\"negativethinmathspace\" /> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{}_r ! F_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for arbitrary <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding=\"application/x-tex\">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-t","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"2 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136017696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrew Lawrie, Jonas Luhrmann, Sung-Jin Oh, Sohrab Shahshahani
We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood–Paley machinery on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {H}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Specifically, some of the estimates established in this paper play a crucial role in the authors’ proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see Lawrie, Lührmann, Oh, and Shahshahani, 2023. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-depen
建立了双曲空间hh上Schrödinger方程的全局时频局部平滑估计mathbb H{^d, d≥2 d }geq 2。在对称的一阶和零阶势的存在下，这些势可能是时间相关的，可能很大，并且有足够快的多项式衰减，这些估计被证明到局部低阶误差。然后，在时间无关的情况下，我们证明了谱条件(即没有阈值共振)意味着在投影到连续谱后的完整局部平滑估计(没有任何误差)。在此过程中，作为频率局域化的手段，我们基于热流原理在h.d mathbb h.d上开发了通用Littlewood-Paley机械。我们的结果和技术的动机是应用在孤立波的稳定性问题上的非线性Schrödinger-type方程{}mathbb H{^}d{。具体地说，本文所建立的一些估计对于证明双曲平面上Schrödinger映射演化下调和映射的非线性渐近稳定性起着至关重要的作用;见Lawrie l hrmann和Shahshahani, 2023。为了证明基于正换向子方法和基于热流的Littlewood-Paley理论的方法的鲁棒性，我们还证明了主要结果在小的时间相关扰动下是稳定的，包括多项式衰减的二阶扰动和小的低阶非对称扰动。}
{"title":"Local Smoothing Estimates for Schrödinger Equations on Hyperbolic Space","authors":"Andrew Lawrie, Jonas Luhrmann, Sung-Jin Oh, Sohrab Shahshahani","doi":"10.1090/memo/1447","DOIUrl":"https://doi.org/10.1090/memo/1447","url":null,"abstract":"We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper H Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">d geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood–Paley machinery on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper H Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {H}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper H Superscript d\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">H</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {H}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Specifically, some of the estimates established in this paper play a crucial role in the authors’ proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see Lawrie, Lührmann, Oh, and Shahshahani, 2023. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-depen","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"217 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius (“Poitou-Tate without restrictions on the order,” 2015) extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.
我们将局部和全局域上有限离散Galois模的经典对偶结果(局部对偶，九项精确序列等)推广到有限类型的所有仿射交换群方案，建立在Česnavičius最近的工作(“Poitou-Tate without restrictions on the order，”2015)的基础上，将这些结果推广到所有有限交换群方案。我们主要集中在较困难的函数域设置上，并在此过程中对数字域的情况做了一些说明。
{"title":"Tate Duality In Positive Dimension over Function Fields","authors":"Zev Rosengarten","doi":"10.1090/memo/1444","DOIUrl":"https://doi.org/10.1090/memo/1444","url":null,"abstract":"We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius (“Poitou-Tate without restrictions on the order,” 2015) extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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