Pub Date : 2020-11-03DOI: 10.1007/978-3-030-89694-2_19
Saeed Nasseh, S. Sather-Wagstaff
{"title":"Applications of Differential Graded Algebra Techniques in Commutative Algebra","authors":"Saeed Nasseh, S. Sather-Wagstaff","doi":"10.1007/978-3-030-89694-2_19","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_19","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89569223","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}
The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski type inequality given in their paper.
{"title":"An inequality in mixed multiplicities","authors":"Suprajo Das","doi":"10.1216/jca.2022.14.509","DOIUrl":"https://doi.org/10.1216/jca.2022.14.509","url":null,"abstract":"The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski type inequality given in their paper.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"8 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84783325","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}
One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős-Renyi-type model for random monomial ideals defined in cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.
{"title":"ASYMPTOTIC DEGREE OF RANDOM MONOMIAL IDEALS","authors":"Lily Silverstein, Dane Wilburne, J. Yang","doi":"10.1216/jca.2023.15.99","DOIUrl":"https://doi.org/10.1216/jca.2023.15.99","url":null,"abstract":"One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős-Renyi-type model for random monomial ideals defined in cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"93 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80261028","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}
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.
{"title":"Dimension of finite free complexes over commutative Noetherian rings","authors":"Lars Christensen, S. Iyengar","doi":"10.1090/conm/773/15529","DOIUrl":"https://doi.org/10.1090/conm/773/15529","url":null,"abstract":"Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"130 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74889816","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":"Poset embeddings of Hilbert functions for two hypersurface rings","authors":"Mitra Koley","doi":"10.1216/jca.2020.12.371","DOIUrl":"https://doi.org/10.1216/jca.2020.12.371","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"68 1","pages":"371-389"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85902030","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":"A characterization of Prüfer $v$-multiplication domains in terms of linear equations","authors":"Lei Qiao, Q. Shu, Fanggui Wang","doi":"10.1216/jca.2020.12.435","DOIUrl":"https://doi.org/10.1216/jca.2020.12.435","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"28 1","pages":"435-445"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86843939","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":"About a variation of local cohomology","authors":"M. A. Khadam, P. Schenzel","doi":"10.1216/jca.2020.12.353","DOIUrl":"https://doi.org/10.1216/jca.2020.12.353","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"12 1","pages":"353-370"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89169354","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}
J. I. García-García, A. Sánchez-R.-Navarro, A. Vigneron-Tenorio
{"title":"Multiple convex body semigroups and Buchsbaum rings","authors":"J. I. García-García, A. Sánchez-R.-Navarro, A. Vigneron-Tenorio","doi":"10.1216/jca.2020.12.309","DOIUrl":"https://doi.org/10.1216/jca.2020.12.309","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"24 1","pages":"309-318"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85038435","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":"The Skolem closure as a semistar operation","authors":"Michaela Steward","doi":"10.1216/jca.2020.12.447","DOIUrl":"https://doi.org/10.1216/jca.2020.12.447","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":"447-457"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83143211","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 an affine domain of dimension n ≥ 3 over a field of characteristic 0. Let L be a projective R[T ]-module of rank 1 and I ⊂ R[T ] a local complete intersection ideal of height n. Assume that I/I is a surjective image of L⊕R[T ]n−1. This paper examines under what conditions I is a surjective image of a projective R[T ]-module P of rank n with determinant L.
{"title":"Projective generation of ideals in polynomial extensions","authors":"M. Keshari, Md. Ali Zinna","doi":"10.1216/jca.2020.12.333","DOIUrl":"https://doi.org/10.1216/jca.2020.12.333","url":null,"abstract":"Let R be an affine domain of dimension n ≥ 3 over a field of characteristic 0. Let L be a projective R[T ]-module of rank 1 and I ⊂ R[T ] a local complete intersection ideal of height n. Assume that I/I is a surjective image of L⊕R[T ]n−1. This paper examines under what conditions I is a surjective image of a projective R[T ]-module P of rank n with determinant L.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"5 1","pages":"333-352"},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83794386","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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