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":"48 1","pages":""},"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":"14 1","pages":""},"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":"67 1","pages":""},"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":"50 1","pages":""},"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":"183 1","pages":""},"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}
Pub Date : 2021-01-24DOI: 10.1007/978-3-030-89694-2_12
Huy Tài Hà, P. Mantero
{"title":"The Alexander–Hirschowitz Theorem and Related Problems","authors":"Huy Tài Hà, P. Mantero","doi":"10.1007/978-3-030-89694-2_12","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_12","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"79 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72584862","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 most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some generalizations of this in the direction of countable CM type by Huneke and Leuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here we consider modules which are high syzygies of MCM modules over non-commutative rings, exploiting the fact that non-commutative rings allow for finer homological behavior. We then generalize Auslander's Theorem in the setting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of global dimension.
{"title":"AUSLANDER’S THEOREM AND N-ISOLATED SINGULARITIES","authors":"Josh Stangle","doi":"10.1216/jca.2023.15.115","DOIUrl":"https://doi.org/10.1216/jca.2023.15.115","url":null,"abstract":"One of the most stunning results in the representation theory of Cohen-Macaulay rings is Auslander's well known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some generalizations of this in the direction of countable CM type by Huneke and Leuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here we consider modules which are high syzygies of MCM modules over non-commutative rings, exploiting the fact that non-commutative rings allow for finer homological behavior. We then generalize Auslander's Theorem in the setting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of global dimension.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91010741","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}
Andrew J. Hetzel, Anna L. Lawson, Andreas Reinhart
In this paper, we advance an ideal-theoretic analogue of a "finite factorization domain" (FFD), giving such a domain the moniker "finite molecularization domain" (FMD). We characterize FMD's as those factorable domains (termed "molecular domains" in the paper) for which every nonzero ideal is divisible by only finitely many nonfactorable ideals (termed "molecules" in the paper) and the monoid of nonzero ideals of the domain is unit-cancellative, in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop a number of connections, particularly at the local level, amongst the concepts of "FMD", "FFD", and the "finite superideal domains" (FSD's) of Hetzel and Lawson. Characterizations of when $k[X^2, X^3]$, where $k$ is a field, and the classical $D+M$ construction are FMD's are provided. We also demonstrate that if $R$ is a Dedekind domain with the finite norm property, then $R[X]$ is an FMD.
{"title":"On finite molecularization domains","authors":"Andrew J. Hetzel, Anna L. Lawson, Andreas Reinhart","doi":"10.1216/JCA.2021.13.69","DOIUrl":"https://doi.org/10.1216/JCA.2021.13.69","url":null,"abstract":"In this paper, we advance an ideal-theoretic analogue of a \"finite factorization domain\" (FFD), giving such a domain the moniker \"finite molecularization domain\" (FMD). We characterize FMD's as those factorable domains (termed \"molecular domains\" in the paper) for which every nonzero ideal is divisible by only finitely many nonfactorable ideals (termed \"molecules\" in the paper) and the monoid of nonzero ideals of the domain is unit-cancellative, in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop a number of connections, particularly at the local level, amongst the concepts of \"FMD\", \"FFD\", and the \"finite superideal domains\" (FSD's) of Hetzel and Lawson. Characterizations of when $k[X^2, X^3]$, where $k$ is a field, and the classical $D+M$ construction are FMD's are provided. We also demonstrate that if $R$ is a Dedekind domain with the finite norm property, then $R[X]$ is an FMD.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87496426","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_25
Adela Vraciu
{"title":"Existence and Constructions of Totally Reflexive Modules","authors":"Adela Vraciu","doi":"10.1007/978-3-030-89694-2_25","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_25","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":"72801270","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_27
V. Welker
{"title":"Which Properties of Stanley–Reisner Rings and Simplicial Complexes are Topological?","authors":"V. Welker","doi":"10.1007/978-3-030-89694-2_27","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_27","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"73 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76466170","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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