Pub Date : 2021-01-01DOI: 10.1007/978-3-030-89694-2_14
B. Harbourne
{"title":"Algebraic Geometry, Commutative Algebra and Combinatorics: Interactions and Open Problems","authors":"B. Harbourne","doi":"10.1007/978-3-030-89694-2_14","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_14","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"141 2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77229206","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_21
B. Olberding
{"title":"The Zariski-Riemann Space of Valuation Rings","authors":"B. Olberding","doi":"10.1007/978-3-030-89694-2_21","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_21","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"70 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78689936","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_16
J. McCullough
{"title":"Subadditivity of Syzygies of Ideals and Related Problems","authors":"J. McCullough","doi":"10.1007/978-3-030-89694-2_16","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_16","url":null,"abstract":"","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":"79654987","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}
Courtney R. Gibbons, David A. Jorgensen, J. Striuli
We introduce a new homological dimension for finitely generated modules over a commutative local ring RR, which is based on a complex derived from a free resolution LL of the residue field of RR, and called LL-dimension. We prove several properties of LL-dimension, give some applications, and compare LL-dimension to complete intersection dimension.
{"title":"𝐿-dimension for modules over a local ring","authors":"Courtney R. Gibbons, David A. Jorgensen, J. Striuli","doi":"10.1090/conm/773/15534","DOIUrl":"https://doi.org/10.1090/conm/773/15534","url":null,"abstract":"<p>We introduce a new homological dimension for finitely generated modules over a commutative 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>, which is based on a complex derived from a free resolution <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the residue field 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>, and called <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension. We prove several properties of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension, give some applications, and compare <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension to complete intersection dimension.</p>","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":"87862281","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_18
N. Minh, Thanh Vu
{"title":"Survey on Regularity of Symbolic Powers of an Edge Ideal","authors":"N. Minh, Thanh Vu","doi":"10.1007/978-3-030-89694-2_18","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_18","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"54 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85737511","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 polynomial ring over a field and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals circled-plus Underscript n Endscripts upper M Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>⨁<!-- ⨁ --></mml:mo> <mml:mi>n</mml:mi> </mml:munder> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">M= bigoplus _n M_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finitely generated graded <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>-module, minimally generated by homogeneous elements of degree zero with a graded <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>-minimal free resolution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbf {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A Cohen-Macaulay module <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> is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e 1"> <mml:semantics> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">e_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the shifts in the graded resolution of <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-t
设R R是一个域上的多项式环，M= φ n M= φ n M= bigo + _n M n是一个有限生成的梯度R R -模，最小由零次齐次元素生成，具有梯度R R -最小自由分辨率F mathbf {F}。当梯度分辨率对称时，Cohen-Macaulay模M M为Gorenstein模。我们给出了第一希尔伯特系数e1e_1的上界，这是根据M M的梯度分辨率的偏移。当M = R/I M = R/I为Gorenstein代数时，该界与[ES09]在准纯分辨的Gorenstein代数中得到的界一致。对于较高的系数，我们推测出一个类似的界。
{"title":"An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules","authors":"Sabine El Khoury, Manoj Kummini, H. Srinivasan","doi":"10.1090/conm/773/15542","DOIUrl":"https://doi.org/10.1090/conm/773/15542","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a polynomial ring over a field and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M equals circled-plus Underscript n Endscripts upper M Subscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:munder>\u0000 <mml:mo>⨁<!-- ⨁ --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:munder>\u0000 <mml:msub>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">M= bigoplus _n M_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a finitely generated graded <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-module, minimally generated by homogeneous elements of degree zero with a graded <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-minimal free resolution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper F\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {F}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. A Cohen-Macaulay module <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">e_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in terms of the shifts in the graded resolution of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-t","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"21 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81322912","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":"On the sum of $z$-ideals in subrings of $C(X)$","authors":"F. Azarpanah, M. Parsinia","doi":"10.1216/jca.2020.12.459","DOIUrl":"https://doi.org/10.1216/jca.2020.12.459","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"6 1","pages":"459-466"},"PeriodicalIF":0.6,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82349076","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 : 2020-11-05DOI: 10.1007/978-3-030-89694-2_3
Nicolás Botbol, A. Dickenstein, H. Schenck
{"title":"The Simplest Minimal Free Resolutions in $${mathbb {P}^1 times mathbb {P}^1}$$","authors":"Nicolás Botbol, A. Dickenstein, H. Schenck","doi":"10.1007/978-3-030-89694-2_3","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_3","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83651814","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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