Notes on endomorphisms, local cohomology and completion

Pub Date : 2021-05-03 DOI:10.1090/conm/773/15540

P. Schenzel

{"title":"Notes on endomorphisms, local cohomology and completion","authors":"P. Schenzel","doi":"10.1090/conm/773/15540","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> denote a finitely generated module over a Noetherian ring <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. For an ideal <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I subset-of upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>I</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:mi>R</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">I \\subset R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> there is a study of the endomorphisms of the local cohomology module <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript upper I Superscript g Baseline left-parenthesis upper M right-parenthesis comma g equals g r a d e left-parenthesis upper I comma upper M right-parenthesis comma\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>H</mml:mi>\n <mml:mi>I</mml:mi>\n <mml:mi>g</mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>g</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>g</mml:mi>\n <mml:mi>r</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:mi>d</mml:mi>\n <mml:mi>e</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>I</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^g_I(M), g = grade(I,M),</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and related results. Another subject is the study of left derived functors of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\n <mml:semantics>\n <mml:mi>I</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-adic completion <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda Subscript i Superscript upper I Baseline left-parenthesis upper H Subscript upper I Superscript g Baseline left-parenthesis upper M right-parenthesis right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi mathvariant=\"normal\">Λ</mml:mi>\n <mml:mi>i</mml:mi>\n <mml:mi>I</mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msubsup>\n <mml:mi>H</mml:mi>\n <mml:mi>I</mml:mi>\n <mml:mi>g</mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>M</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Lambda ^I_i(H^g_I(M))</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, motivated by a characterization of Gorenstein rings given in <bold>[25]</bold>. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/conm/773/15540","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 3

Abstract

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

查看原文

关于自同态、局部上同调和补全的注解

设M M表示诺瑟环rr上有限生成的模。对于理想I∧R I \子集R，研究了局部上同模H I g (M)， g = g R ade(I,M)， H^g_I(M)， g = grade(I,M)的自同态及其相关结果。另一个主题是研究I - I进补全Λ I I(H I g (M)) \Lambda ^I_i(H^g_I(M))的左衍生函子，其动机是在[25]中给出的Gorenstein环的表征。这提供了另一个科恩-麦考利标准。通过几个实例说明了结果。对于两个不同的局部上同模的同态也有一个推广。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文