Pub Date : 2021-06-15DOI: 10.1007/978-3-030-89694-2_15
S. Iyengar, Linquan Ma, Karl Schwede, M. Walker
{"title":"Maximal Cohen-Macaulay Complexes and Their Uses: A Partial Survey","authors":"S. Iyengar, Linquan Ma, Karl Schwede, M. Walker","doi":"10.1007/978-3-030-89694-2_15","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_15","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89842573","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 : 2021-06-08DOI: 10.1007/978-3-030-89694-2_23
Claudiu Raicu, Steven V. Sam
{"title":"Hermite Reciprocity and Schwarzenberger Bundles","authors":"Claudiu Raicu, Steven V. Sam","doi":"10.1007/978-3-030-89694-2_23","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_23","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"74 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84411302","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}
It was shown by Kang (1989) that a domain R is a Krull domain if and only if R is a Mori domain and a PvMD. In this paper, we extend this result to Gorenstein multiplicative ideal theory. To do this, we introduce the concepts of FT-domains and G-PvMDs, and study them by a new star-operation, i.e., the f-operation. We prove that (1) a domain R is an integrally closed FT-domain if and only if R is a P-domain; (2) a domain R is a G-PvMD if and only if R is a g-coherent FT-domain; (3) a domain R is a G-Krull domain if and only if R is a Mori domain and a G-Pv$v$MD.
{"title":"FT-domains and Gorenstein Prüfer v-multiplication domains","authors":"Shiqi Xing","doi":"10.1216/jca.2021.13.263","DOIUrl":"https://doi.org/10.1216/jca.2021.13.263","url":null,"abstract":"It was shown by Kang (1989) that a domain R is a Krull domain if and only if R is a Mori domain and a PvMD. In this paper, we extend this result to Gorenstein multiplicative ideal theory. To do this, we introduce the concepts of FT-domains and G-PvMDs, and study them by a new star-operation, i.e., the f-operation. We prove that (1) a domain R is an integrally closed FT-domain if and only if R is a P-domain; (2) a domain R is a G-PvMD if and only if R is a g-coherent FT-domain; (3) a domain R is a G-Krull domain if and only if R is a Mori domain and a G-Pv$v$MD.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"157 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73739476","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 a polarized variety (X,D), we can associate a graded ring and a Hilbert series. Assume D is an ample ℚ Cartier divisor, and (X,D) is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension ≤1, that is, both singularities of dimension 1 and singular points, extending a 2013 result of Buckley, Reid and the author about varieties with only isolated singularities.
{"title":"Dedekind sums and parsing of Hilbert series","authors":"Shengtian Zhou","doi":"10.1216/jca.2021.13.281","DOIUrl":"https://doi.org/10.1216/jca.2021.13.281","url":null,"abstract":"Given a polarized variety (X,D), we can associate a graded ring and a Hilbert series. Assume D is an ample ℚ Cartier divisor, and (X,D) is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension ≤1, that is, both singularities of dimension 1 and singular points, extending a 2013 result of Buckley, Reid and the author about varieties with only isolated singularities.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"304 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82878491","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}
Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miro-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.
确定弱Lefschetz性质的存在通常是一个具有挑战性的问题。在Brenner and Kaid(2007)、Cook II and Nagel(2011)以及Migliore、micro - roig、Murai and Nagel(2013)的研究基础上，我们对三个变量中的四个发生器的Artinian单项式理想进行了深入研究。我们利用菱形拼接的连接描述了这种理想的合束的半稳定性，确定了它的一般分裂类型，并确定了弱Lefschetz性质的存在。我们给出了特征零和正特征的结果。
{"title":"Syzygy bundles and the weak Lefschetz property of monomial almost complete intersections","authors":"D. Cook, U. Nagel","doi":"10.1216/jca.2021.13.157","DOIUrl":"https://doi.org/10.1216/jca.2021.13.157","url":null,"abstract":"Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miro-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"71 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84205205","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}
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":"19 2","pages":""},"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":"6 1","pages":""},"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":"36 1","pages":""},"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}
<p>Let <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> denote a finitely generated module over a Noetherian 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>. For an ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I subset-of upper R"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">I subset R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a study of the endomorphisms of the local cohomology module <inline-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>I</mml:mi> <mml:mi>g</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mi>r</mml:mi> <mml:mi>a</mml:mi> <mml:mi>d</mml:mi> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^g_I(M), g = grade(I,M),</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and related results. Another subject is the study of left derived functors of the <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>-adic completion <inline-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi
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":"18 1","pages":""},"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":"29 1","pages":""},"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}
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