E. Guardo, M. Kreuzer, Tran N. K. Linh, L. N. Long
Let 𝕏 be a set of K-rational points in ℙ1×ℙ1 over a field K of characteristic zero, let 𝕐 be a fat point scheme supported at 𝕏, and let R𝕐 be the bihomogeneous coordinate ring of 𝕐. In this paper we investigate the module of Kahler differentials ΩR𝕐∕K1. We describe this bigraded R𝕐-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support 𝕏 is a complete intersection or an almost complete intersection in ℙ1×ℙ1. Moreover, we introduce a Kahler different for 𝕐 and use it to characterize ACM reduced schemes in ℙ1×ℙ1 having the Cayley–Bacharach property.
{"title":"Kähler Differentials for Fat Point Schemes in ℙ1×ℙ1","authors":"E. Guardo, M. Kreuzer, Tran N. K. Linh, L. N. Long","doi":"10.1216/jca.2021.13.179","DOIUrl":"https://doi.org/10.1216/jca.2021.13.179","url":null,"abstract":"Let 𝕏 be a set of K-rational points in ℙ1×ℙ1 over a field K of characteristic zero, let 𝕐 be a fat point scheme supported at 𝕏, and let R𝕐 be the bihomogeneous coordinate ring of 𝕐. In this paper we investigate the module of Kahler differentials ΩR𝕐∕K1. We describe this bigraded R𝕐-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support 𝕏 is a complete intersection or an almost complete intersection in ℙ1×ℙ1. Moreover, we introduce a Kahler different for 𝕐 and use it to characterize ACM reduced schemes in ℙ1×ℙ1 having the Cayley–Bacharach property.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72580621","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 develop a local cohomology theory for ℱℐm-modules, and show that it in many ways mimics the classical theory for multigraded modules over a polynomial ring. In particular, we define an invariant of ℱℐm-modules using this local cohomology theory which closely resembles an invariant of multigraded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
{"title":"Local cohomology and the multigraded regularity of ℱℐm-modules","authors":"Liping Li, Eric Ramos","doi":"10.1216/jca.2021.13.235","DOIUrl":"https://doi.org/10.1216/jca.2021.13.235","url":null,"abstract":"We develop a local cohomology theory for ℱℐm-modules, and show that it in many ways mimics the classical theory for multigraded modules over a polynomial ring. In particular, we define an invariant of ℱℐm-modules using this local cohomology theory which closely resembles an invariant of multigraded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90516694","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}
Boij–Soderberg theory describes the Betti diagrams of graded modules over a polynomial ring as a linear combination of pure diagrams with positive coefficients. In this paper, we focus on the Betti diagrams of lexicographic ideals. Mainly, we characterize the Boij–Soderberg decomposition of the Betti table of a lexicographic ideal in the polynomial ring with three variables, and show a nice connection between its Boij–Soderberg decomposition and the ones of other related lexicographic ideals.
{"title":"Boij–Söderberg decompositions of lexicographic ideals","authors":"Sema Güntürkün","doi":"10.1216/jca.2021.13.209","DOIUrl":"https://doi.org/10.1216/jca.2021.13.209","url":null,"abstract":"Boij–Soderberg theory describes the Betti diagrams of graded modules over a polynomial ring as a linear combination of pure diagrams with positive coefficients. In this paper, we focus on the Betti diagrams of lexicographic ideals. Mainly, we characterize the Boij–Soderberg decomposition of the Betti table of a lexicographic ideal in the polynomial ring with three variables, and show a nice connection between its Boij–Soderberg decomposition and the ones of other related lexicographic ideals.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90251826","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}
Let MM denote a finitely generated module over a Noetherian ring RR. For an ideal I⊂RI subset R there is a study of the endomorphisms of the local cohomology module HIg(M),g=grade(I,M),H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the II-adic completion Λ
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。
{"title":"Notes on endomorphisms, local cohomology and completion","authors":"P. Schenzel","doi":"10.1090/conm/773/15540","DOIUrl":"https://doi.org/10.1090/conm/773/15540","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 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> denote a finitely generated module over a Noetherian ring <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>. For an ideal <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I subset-of upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mi>R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">I subset R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> there is a study of the endomorphisms of the local cohomology module <inline-formula content-type=\"math/mathml\">\u0000<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\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mi>g</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mi>a</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^g_I(M), g = grade(I,M),</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and related results. Another subject is the study of left derived functors of the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic completion <inline-formula content-type=\"math/mathml\">\u0000<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\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mi","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80948324","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}
Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r geq max{2,d+1}$. Then the action of $E(A[M] oplus P)$ on $Um(A[M] oplus P)$ is transitive and (2) Assume $(R, m, K)$ is a regular local ring containing a field $k$ such that either $char$ $k=0$ or $ char$ $k = p$ and $tr$-$deg$ $K/mathbb{F}_p geq 1$. Let $A$ be a ring of type $R[d,m,n]^*$ and $fin R$ be a regular parameter. Then all finitely generated projective modules over $A[M],$ $A[M]_f$ and $A[M] otimes_R R(T)$ are free. When $M$ is free both results are due to Keshari and Lokhande.
{"title":"Unimodular rows over monoid extensions of overrings of polynomial rings","authors":"M. A. Mathew, M. Keshari","doi":"10.1216/jca.2022.14.583","DOIUrl":"https://doi.org/10.1216/jca.2022.14.583","url":null,"abstract":"Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r geq max{2,d+1}$. Then the action of $E(A[M] oplus P)$ on $Um(A[M] oplus P)$ is transitive and (2) Assume $(R, m, K)$ is a regular local ring containing a field $k$ such that either $char$ $k=0$ or $ char$ $k = p$ and $tr$-$deg$ $K/mathbb{F}_p geq 1$. Let $A$ be a ring of type $R[d,m,n]^*$ and $fin R$ be a regular parameter. Then all finitely generated projective modules over $A[M],$ $A[M]_f$ and $A[M] otimes_R R(T)$ are free. When $M$ is free both results are due to Keshari and Lokhande.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83322544","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}
Given that symbolic and ordinary powers of an ideal do not always coincide, we look for conditions on the ideal such that equality holds for every natural number. This paper focuses on studying the equality for Derksen ideals defined by finite groups acting linearly on a polynomial ring.
{"title":"SYMBOLIC POWERS OF DERKSEN IDEALS","authors":"Sandra Sandoval-G'omez","doi":"10.1216/jca.2023.15.275","DOIUrl":"https://doi.org/10.1216/jca.2023.15.275","url":null,"abstract":"Given that symbolic and ordinary powers of an ideal do not always coincide, we look for conditions on the ideal such that equality holds for every natural number. This paper focuses on studying the equality for Derksen ideals defined by finite groups acting linearly on a polynomial ring.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76281783","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}
Let $R$ be a commutative ring with identity. The small finitistic dimension $fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $fPD(R)leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $fPD(R)= sup{grade(m,R)|min Max(R)}$ where $grade(m,R)$ is the grade of $m$ on $R$ . We also show that a ring $R$ satisfies $fPD(R)leq 1$ if and only if $R$ is a $DW$ ring. As applications, we show that the small finitistic dimensions of strong Prufer rings and $LPVD$s are at most one. Moreover, for any given $nin mathbb{N}$, we obtain examples of total rings of quotients $R$ with $fPD(R)=n$.
{"title":"THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS","authors":"Xiaolei Zhang, Fanggui Wang","doi":"10.1216/jca.2023.15.131","DOIUrl":"https://doi.org/10.1216/jca.2023.15.131","url":null,"abstract":"Let $R$ be a commutative ring with identity. The small finitistic dimension $fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $fPD(R)leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $fPD(R)= sup{grade(m,R)|min Max(R)}$ where $grade(m,R)$ is the grade of $m$ on $R$ . We also show that a ring $R$ satisfies $fPD(R)leq 1$ if and only if $R$ is a $DW$ ring. As applications, we show that the small finitistic dimensions of strong Prufer rings and $LPVD$s are at most one. Moreover, for any given $nin mathbb{N}$, we obtain examples of total rings of quotients $R$ with $fPD(R)=n$.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82894095","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":"Discriminant amoebas and lopsidedness","authors":"Jens Forsgård","doi":"10.1216/JCA.2021.13.41","DOIUrl":"https://doi.org/10.1216/JCA.2021.13.41","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85782531","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":"Fat point ideals in $mathbb{K}[mathbb{P}^N]$ with linear minimal free resolutions and their resurgences","authors":"Hassan Haghighi, M. Mosakhani","doi":"10.1216/JCA.2021.13.61","DOIUrl":"https://doi.org/10.1216/JCA.2021.13.61","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85635594","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":"Root extension in polynomial and power series rings","authors":"Mi Hee Park","doi":"10.1216/JCA.2021.13.129","DOIUrl":"https://doi.org/10.1216/JCA.2021.13.129","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74626724","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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