Pub Date : 2021-06-28DOI: 10.1007/978-3-030-89694-2_5
G. Caviglia, Alessandro De Stefani, E. Sbarra
{"title":"The Eisenbud-Green-Harris Conjecture","authors":"G. Caviglia, Alessandro De Stefani, E. Sbarra","doi":"10.1007/978-3-030-89694-2_5","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_5","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82950669","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 ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0rightarrow mathrm{Hom}_R(P,A)rightarrow mathrm{Hom}_R(P,B)rightarrow mathrm{Hom}_R(P,C)rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.
{"title":"CHARACTERIZING S-PROJECTIVE MODULES AND S-SEMISIMPLE RINGS BY UNIFORMITY","authors":"Xiaolei Zhang, W. Qi","doi":"10.1216/jca.2023.15.139","DOIUrl":"https://doi.org/10.1216/jca.2023.15.139","url":null,"abstract":"Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0rightarrow mathrm{Hom}_R(P,A)rightarrow mathrm{Hom}_R(P,B)rightarrow mathrm{Hom}_R(P,C)rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73368745","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 : 2021-06-16DOI: 10.1007/978-3-030-89694-2_1
Josep Àlvarez Montaner, J. Jeffries, Luis N'unez-Betancourt
{"title":"Bernstein-Sato Polynomials in Commutative Algebra","authors":"Josep Àlvarez Montaner, J. Jeffries, Luis N'unez-Betancourt","doi":"10.1007/978-3-030-89694-2_1","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_1","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73948435","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 : 2021-06-15DOI: 10.1007/978-3-030-89694-2_13
Huy Tài Hà, N. Trung
{"title":"Depth Functions and Symbolic Depth Functions of Homogeneous Ideals","authors":"Huy Tài Hà, N. Trung","doi":"10.1007/978-3-030-89694-2_13","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_13","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82320718","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 : 2021-06-15DOI: 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":null,"pages":null},"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":null,"pages":null},"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":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":"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}
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":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":"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}
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":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":"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}
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