{"title":"交换诺特环上的大纯投影模块:与完形比较","authors":"Dolors Herbera, Pavel Příhoda, Roger Wiegand","doi":"10.1515/forum-2024-0031","DOIUrl":null,"url":null,"abstract":"A module over a ring <jats:italic>R</jats:italic> is <jats:italic>pure projective</jats:italic> provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module <jats:italic>M</jats:italic>, we consider <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_1297.png\"/> <jats:tex-math>{\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which consists of direct summands of direct sums of copies of <jats:italic>M</jats:italic>. We are primarily interested in the case where <jats:italic>R</jats:italic> is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_1297.png\"/> <jats:tex-math>{\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of <jats:italic>M</jats:italic> and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_0807.png\"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of isomorphism classes of countably generated modules in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_1297.png\"/> <jats:tex-math>{\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with the addition induced by the direct sum. We show that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_0807.png\"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a submonoid of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>M</m:mi> <m:msub> <m:mo>⊗</m:mo> <m:mi>R</m:mi> </m:msub> <m:mover accent=\"true\"> <m:mi>R</m:mi> <m:mo>^</m:mo> </m:mover> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2024-0031_eq_0808.png\"/> <jats:tex-math>{V^{*}(M\\otimes_{R}\\widehat{R})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, this allows us to make computations with examples and to prove some realization results.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Big pure projective modules over commutative noetherian rings: Comparison with the completion\",\"authors\":\"Dolors Herbera, Pavel Příhoda, Roger Wiegand\",\"doi\":\"10.1515/forum-2024-0031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A module over a ring <jats:italic>R</jats:italic> is <jats:italic>pure projective</jats:italic> provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module <jats:italic>M</jats:italic>, we consider <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_1297.png\\\"/> <jats:tex-math>{\\\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which consists of direct summands of direct sums of copies of <jats:italic>M</jats:italic>. We are primarily interested in the case where <jats:italic>R</jats:italic> is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_1297.png\\\"/> <jats:tex-math>{\\\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of <jats:italic>M</jats:italic> and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_0807.png\\\"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of isomorphism classes of countably generated modules in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>Add</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_1297.png\\\"/> <jats:tex-math>{\\\\operatorname{Add}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with the addition induced by the direct sum. We show that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_0807.png\\\"/> <jats:tex-math>{V^{*}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a submonoid of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>V</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>M</m:mi> <m:msub> <m:mo>⊗</m:mo> <m:mi>R</m:mi> </m:msub> <m:mover accent=\\\"true\\\"> <m:mi>R</m:mi> <m:mo>^</m:mo> </m:mover> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2024-0031_eq_0808.png\\\"/> <jats:tex-math>{V^{*}(M\\\\otimes_{R}\\\\widehat{R})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, this allows us to make computations with examples and to prove some realization results.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2024-0031\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2024-0031","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
环 R 上的模块是纯投影模块,前提是它与有限呈现模块的直接和的直接和同构。我们开发了对交换诺特环上的纯投影模块进行分类的工具。特别是,对于一个固定的有限呈现模块 M,我们考虑 Add ( M ) {\operatorname{Add}(M)} ,它由 M 的副本的直接和的直接和组成。我们主要关注 R 是一维局部域的情况,以及无扭(或 Cohen-Macaulay)模块。我们证明,即使在这种情况下,Add ( M ) {\operatorname{Add}(M)} 也可以有大量模块不是有限生成模块的直和。我们的工作基于这样一个事实:这种无限生成的直接和都是由有限生成的数据决定的。也就是说,M 的内态环的幂幂/迹理想和有限生成的投影模块模都是由这些幂幂理想决定的。这样,我们就可以将研究有限生成模块的直接和分解与其完备性比较的经典理论扩展到无限生成的情况。我们研究了单元 V * ( M ) {V^{*}(M)} 的结构,即 Add ( M ) {\operatorname{Add}(M)} 中由直接相加诱导的可数生成模块的同构类。我们证明了 V * ( M ) {V^{*}(M)} 是 V * ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} 的子单体,这让我们可以进行计算。 这样,我们就可以利用实例进行计算,并证明一些实现结果。
Big pure projective modules over commutative noetherian rings: Comparison with the completion
A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add(M){\operatorname{Add}(M)}, which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add(M){\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V*(M){V^{*}(M)}, of isomorphism classes of countably generated modules in Add(M){\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V*(M){V^{*}(M)} is a submonoid of V*(M⊗RR^){V^{*}(M\otimes_{R}\widehat{R})}, this allows us to make computations with examples and to prove some realization results.
期刊介绍:
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.