Sunil K. Chebolu, J. Corry, Elizabeth Grimm, Andrew Hatfield
{"title":"RINGS WITH AN ELEMENTARY ABELIAN p-GROUP OF UNITS","authors":"Sunil K. Chebolu, J. Corry, Elizabeth Grimm, Andrew Hatfield","doi":"10.1216/jca.2023.15.469","DOIUrl":"https://doi.org/10.1216/jca.2023.15.469","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138986457","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":"ON ABELIAN GROUPS HAVING ISOMORPHIC PROPER CHARACTERISTIC SUBGROUPS","authors":"A. Chekhlov, P. Danchev","doi":"10.1216/jca.2023.15.481","DOIUrl":"https://doi.org/10.1216/jca.2023.15.481","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139017679","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 Tjurina ideal of a germ of an holomorphic function $f$ is the ideal of $mathscr{O}_{mathbbm{C}^n,0}$ - the ring of those germs at $0inmathbbm{C}^n$ - generated by $f$ itself and by its partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $(mathbbm{C}^n,0)$ to the hypersurface singularity defined by $f$, being an object of central interest in Singularity Theory. In this note we introduce emph{$T$-fullness} and emph{$T$-dependence}, two easily verifiable properties for arbitrary ideals of germs of holomorphic functions. These two properties allow us to give necessary and sufficient conditions on an ideal $Isubset mathscr{O}_{mathbbm{C}^n,0}$, for the equation $I=T(f)$ to admit a solution $f$.
{"title":"ON TJURINA IDEALS OF HYPERSURFACE SINGULARITIES","authors":"João Hélder Olmedo Rodrigues","doi":"10.1216/jca.2023.15.261","DOIUrl":"https://doi.org/10.1216/jca.2023.15.261","url":null,"abstract":"The Tjurina ideal of a germ of an holomorphic function $f$ is the ideal of $mathscr{O}_{mathbbm{C}^n,0}$ - the ring of those germs at $0inmathbbm{C}^n$ - generated by $f$ itself and by its partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $(mathbbm{C}^n,0)$ to the hypersurface singularity defined by $f$, being an object of central interest in Singularity Theory. In this note we introduce emph{$T$-fullness} and emph{$T$-dependence}, two easily verifiable properties for arbitrary ideals of germs of holomorphic functions. These two properties allow us to give necessary and sufficient conditions on an ideal $Isubset mathscr{O}_{mathbbm{C}^n,0}$, for the equation $I=T(f)$ to admit a solution $f$.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82575875","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":"CHARACTERIZATION OF FERMAT HYPERSURFACES BY ABELIAN GROUPS","authors":"Taro Hayashi","doi":"10.1216/jca.2023.15.233","DOIUrl":"https://doi.org/10.1216/jca.2023.15.233","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76502156","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}
A BSTRACT . A commutative ring with unity is called an N -ring if each of its ideals is contracted from an Noetherian extension ring. The chief result of the paper is a characterisation of local N -rings by their subdirectly irreducible quotients. The results are applied to characterise spectral synthesis on G -invariant subspaces of the space of complex valued functions on an Abelian group G .
{"title":"A CHARACTERISATION OF LOCAL N-RINGS AND AN APPLICATION TO ABSTRACT HARMONIC ANALYSIS","authors":"B. Wilkens","doi":"10.1216/jca.2023.15.287","DOIUrl":"https://doi.org/10.1216/jca.2023.15.287","url":null,"abstract":"A BSTRACT . A commutative ring with unity is called an N -ring if each of its ideals is contracted from an Noetherian extension ring. The chief result of the paper is a characterisation of local N -rings by their subdirectly irreducible quotients. The results are applied to characterise spectral synthesis on G -invariant subspaces of the space of complex valued functions on an Abelian group G .","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85507345","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 reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points Z in P2, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement AZ . Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set Z has the cardinality equal to 11 or 12, and describe five cases where this happens.
{"title":"UNEXPECTED CURVES IN ℙ2, LINE ARRANGEMENTS, AND MINIMAL DEGREE OF JACOBIAN RELATIONS","authors":"A. Dimca","doi":"10.1216/jca.2023.15.15","DOIUrl":"https://doi.org/10.1216/jca.2023.15.15","url":null,"abstract":"We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points Z in P2, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement AZ . Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set Z has the cardinality equal to 11 or 12, and describe five cases where this happens.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86750277","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 consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.
{"title":"TORSION-FREE EXTENSIONS OF PROJECTIVE MODULES BY TORSION MODULES","authors":"L. Fuchs","doi":"10.1216/jca.2023.15.31","DOIUrl":"https://doi.org/10.1216/jca.2023.15.31","url":null,"abstract":"We consider a generalization of a problem raised by P. Griffith [12] on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin [2]. Torsion modules T with the following property are characterized: if M is a torsion-free module and F is a projective submodule such that M/F ∼= T , then M is projective (Theorem 4.1). It is shown in Theorem 6.4 that for abelian groups whose cardinality is not cofinal with ω this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy [9]. The problem for valuation domains is also discussed, the results are similar to the case of abelian groups.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82859113","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, m ) be a local ring, Z a specialization closed subset of Spec R and X an R -complex with finitely generated homology and finite dimension. We show that Att R H dim X Z ( X ) = { p ∈ Supp R X | cd( Z , R/ p ) − inf X p = dim R X } . We also represent a generalization of Lichtenbaum-Hartshorne Vanishing Theorem for complexes of R -modules.
． 设(R, m)是一个局部环，Z是Spec R和X的专门化闭子集，是一个具有有限生成同调和有限维数的R -复形。我们证明了Att R H dim X Z (X) = {p∈Supp R X | cd(Z, R/ p)−inf X p = dim R X}。我们还对R -模复形的Lichtenbaum-Hartshorne消失定理进行了推广。
{"title":"ATTACHED PRIMES OF LOCAL COHOMOLOGY MODULES OF COMPLEXES","authors":"Nguyen Minh Tri","doi":"10.1216/jca.2023.15.75","DOIUrl":"https://doi.org/10.1216/jca.2023.15.75","url":null,"abstract":". Let ( R, m ) be a local ring, Z a specialization closed subset of Spec R and X an R -complex with finitely generated homology and finite dimension. We show that Att R H dim X Z ( X ) = { p ∈ Supp R X | cd( Z , R/ p ) − inf X p = dim R X } . We also represent a generalization of Lichtenbaum-Hartshorne Vanishing Theorem for complexes of R -modules.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86007676","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}
. Unexpected hypersurface is a name given an element to some particular linear system introduced around in [2], motivated by work of Di Gennaro, Ilardi and Vall`es and of Faenzi and Vall`es, and it is a field of great study since then. It attracts many people because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties, etc. In this paper we continue the study about BMSS duality, of [3]. In [6], the authors introduced the concept of unexpected hypersurfaces and they explain the so-called BMSS duality showing that unexpected curves are in some sense dual to their tangent cones at their singular point. In this paper, we revisit the configuration of points associated to Hesse arrangement, Hesse union dual Hesse arrangement and we study the geometry of the associated varieties and their companions.
{"title":"COMPANION VARIETIES FOR HESSE, HESSE UNION DUAL HESSE ARRANGEMENTS","authors":"Pietro De Poi, G. Ilardi","doi":"10.1216/jca.2023.15.1","DOIUrl":"https://doi.org/10.1216/jca.2023.15.1","url":null,"abstract":". Unexpected hypersurface is a name given an element to some particular linear system introduced around in [2], motivated by work of Di Gennaro, Ilardi and Vall`es and of Faenzi and Vall`es, and it is a field of great study since then. It attracts many people because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties, etc. In this paper we continue the study about BMSS duality, of [3]. In [6], the authors introduced the concept of unexpected hypersurfaces and they explain the so-called BMSS duality showing that unexpected curves are in some sense dual to their tangent cones at their singular point. In this paper, we revisit the configuration of points associated to Hesse arrangement, Hesse union dual Hesse arrangement and we study the geometry of the associated varieties and their companions.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83889256","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":"REDUCTIONS OF IDEALS IN PRÜFER RINGS","authors":"M. Jarrar, S. Kabbaj","doi":"10.1216/jca.2023.15.45","DOIUrl":"https://doi.org/10.1216/jca.2023.15.45","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89542413","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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