In this paper we give a generalization of the coefficient ideals of an mmathfrak {m}-primary ideal II in a quasi-unmixed local ring RR with infinite residue field.
{"title":"A generalization of coefficient ideals","authors":"P. Lima","doi":"10.1090/conm/773/15537","DOIUrl":"https://doi.org/10.1090/conm/773/15537","url":null,"abstract":"<p>In this paper we give a generalization of the coefficient ideals of an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German m\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">m</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak {m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-primary ideal <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=\"application/x-tex\">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a quasi-unmixed local ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with infinite residue field.</p>","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78850280","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_17
J. Migliore, U. Nagel
{"title":"Applications of Liaison","authors":"J. Migliore, U. Nagel","doi":"10.1007/978-3-030-89694-2_17","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_17","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"35 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75600910","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>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complete local ring and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime ideal of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is determined precisely which conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are equivalent to the existence of a complete unramified regular local ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g element-of upper A minus upper Q"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">gin A-Q</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 R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript g Baseline long right-arrow upper R Subscript g"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy="false">⟶<!-- ⟶ --></mml:mo>
设R R是一个完全局部环，Q Q是R R的素理想。精确地确定了在R R上哪些条件等价于存在一个完全的非发散正则局部环a a和a -Q中的一个元素g∈a−Q g，使得R R是一个有限的a a -模，并且a g R R_g。在此过程中发展了可能嵌入A × R的许多其他性质，包括确定Cohen-Gabber定理中哪些场可以是系数场。
{"title":"The Étale locus in complete local rings","authors":"R. Heitmann","doi":"10.1090/conm/773/15532","DOIUrl":"https://doi.org/10.1090/conm/773/15532","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complete local ring and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q\"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding=\"application/x-tex\">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime ideal of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is determined precisely which conditions on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are equivalent to the existence of a complete unramified regular local ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an element <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g element-of upper A minus upper Q\"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>A</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">gin A-Q</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 R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript g Baseline long right-arrow upper R Subscript g\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">⟶<!-- ⟶ --></mml:mo> ","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"137 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79735692","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}
{"title":"Magic squares of squares over a finite field","authors":"S. Hengeveld, Giancarlo Labruna, Aihua Li","doi":"10.1090/conm/773/15536","DOIUrl":"https://doi.org/10.1090/conm/773/15536","url":null,"abstract":"<p>A magic square <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over an integral domain <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3 times 3\"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">3times 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix with entries from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the elements from each row, column, and diagonal add to the same sum. If all the entries in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are perfect squares in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we call <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a magic square of squares over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=\"application/x-tex\">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Z}</mml:annotation> ","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"39 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85803885","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_9
S. Cutkosky, H. Srinivasan
{"title":"Multiplicities and Mixed Multiplicities of Filtrations","authors":"S. Cutkosky, H. Srinivasan","doi":"10.1007/978-3-030-89694-2_9","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_9","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91195487","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_24
Ryo Takahashi
{"title":"Generation in Module Categories and Derived Categories of Commutative Rings","authors":"Ryo Takahashi","doi":"10.1007/978-3-030-89694-2_24","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_24","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"47 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85583717","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}
In the 1960’s, Matlis defined an h h -local domain to be a (commutative) integral domain in which each nonzero element is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. For rings with zero-divisors, by changing “nonzero” to “regular,” one obtains the definition of an h h -local ring. Nearly two dozen equivalent characterizations of h h -local domain have appeared in the literature. We show that most of these remain equivalent to h h -local ring if one also replaces “localization” by “regular localization” and assumes that the ring is a Marot ring (i.e., every regular ideal is generated by its regular elements).
在20世纪60年代，Matlis将h局部域定义为一个(交换)积分域，其中每个非零元素只包含在有限多个极大理想中，每个非零素数理想包含在一个唯一的极大理想中。对于具有零因子的环，通过将“非零”变为“正则”，可以得到h - h局部环的定义。在文献中出现了近24种h - h局部域的等效表征。我们证明，如果用“正则局部化”代替“局部化”，并假设环是一个Marot环(即，每个正则理想都是由它的正则元素生成的)，这些环中的大多数仍然等价于h - h -局部环。
{"title":"ℎ-local Rings","authors":"L. Klingler, A. Omairi","doi":"10.1090/conm/773/15535","DOIUrl":"https://doi.org/10.1090/conm/773/15535","url":null,"abstract":"In the 1960’s, Matlis defined an h h -local domain to be a (commutative) integral domain in which each nonzero element is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. For rings with zero-divisors, by changing “nonzero” to “regular,” one obtains the definition of an h h -local ring. Nearly two dozen equivalent characterizations of h h -local domain have appeared in the literature. We show that most of these remain equivalent to h h -local ring if one also replaces “localization” by “regular localization” and assumes that the ring is a Marot ring (i.e., every regular ideal is generated by its regular elements).","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"37 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84245660","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_20
Wenbo Niu
{"title":"Regularity Bounds by Projection","authors":"Wenbo Niu","doi":"10.1007/978-3-030-89694-2_20","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_20","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"65 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86071803","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_22
D. Patil, J. K. Verma
{"title":"Rational Points and Trace Forms on a Finite Algebra over a Real Closed Field","authors":"D. Patil, J. K. Verma","doi":"10.1007/978-3-030-89694-2_22","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_22","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90094002","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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