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":"90 1","pages":""},"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":"21 1","pages":""},"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":"39 1","pages":""},"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}
{"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":"94 1","pages":""},"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}
{"title":"GENERATING POLYNOMIALS FOR THE DISTRIBUTION OF GENERALIZED BINOMIAL COEFFICIENTS IN DISCRETE VALUATION DOMAINS","authors":"Dong Quan Ngoc Nguyen","doi":"10.1216/jca.2023.15.65","DOIUrl":"https://doi.org/10.1216/jca.2023.15.65","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"53 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76909324","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":"167 1","pages":""},"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}
. Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen-Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.
{"title":"Generalized power series with a limited number of factorizations","authors":"Ngoc P. Aylesworth, J. R. Juett","doi":"10.1216/jca.2022.14.471","DOIUrl":"https://doi.org/10.1216/jca.2022.14.471","url":null,"abstract":". Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen-Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"24 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87238206","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 study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Having an inspiration in the representation theory of orders in separable algebras we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.
{"title":"Generalized lattices over one-dimensional noetherian domains","authors":"P. Př́ıhoda","doi":"10.1216/jca.2022.14.443","DOIUrl":"https://doi.org/10.1216/jca.2022.14.443","url":null,"abstract":"We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Having an inspiration in the representation theory of orders in separable algebras we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81896862","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 new class of abelian p-groups is introduced, the countably totally projective groups, that contains the well-known class of totally projective groups. A countably totally projective group is shown to have the property that every fully inert subgroup is commensurable with a fully invariant subgroup. This generalizes results of Goldsmith, Salce and Zanardo (2014), who proved that a direct sum of cyclic p-groups has this property. It also answers affirmatively two questions recently posed in the literature.
{"title":"Countably totally projective Abelian p-groups have minimal full inertia","authors":"P. Keef","doi":"10.1216/jca.2022.14.427","DOIUrl":"https://doi.org/10.1216/jca.2022.14.427","url":null,"abstract":"A new class of abelian p-groups is introduced, the countably totally projective groups, that contains the well-known class of totally projective groups. A countably totally projective group is shown to have the property that every fully inert subgroup is commensurable with a fully invariant subgroup. This generalizes results of Goldsmith, Salce and Zanardo (2014), who proved that a direct sum of cyclic p-groups has this property. It also answers affirmatively two questions recently posed in the literature.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"25 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72815220","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 collection of t subspaces in an ndimensional vector space W we can associate to them t linear ideals in the symmetric algebra S(W ∗). Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t. In this paper we show that analogous results hold when we work over the exterior algebra ∧ (W ∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.
给定一个n维向量空间W中的t个子空间的集合，我们可以将对称代数S(W *)中的t个线性理想与它们联系起来。孔卡和赫尔佐格显示产品的Castelnuovo-Mumford规律的线性理想= t t。Derksen和Sidman表明Castelnuovo-Mumford规律的交集t线性理想是最多t。在本文中,我们表明,类似的结果持有当我们工作外代数∧(W∗)(0)特征的领域。为了证明这些结果我们依靠functoriality等变化自由决议和构造一个函子Ω的范畴自身的多项式函子。函子Ω将与对称代数上的子空间排列相关的多项式函子的分辨率转换为外部代数上的分辨率。
{"title":"Resolutions of ideals of subspace arrangements","authors":"Francesca Gandini","doi":"10.1216/jca.2022.14.319","DOIUrl":"https://doi.org/10.1216/jca.2022.14.319","url":null,"abstract":"Given a collection of t subspaces in an ndimensional vector space W we can associate to them t linear ideals in the symmetric algebra S(W ∗). Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t. In this paper we show that analogous results hold when we work over the exterior algebra ∧ (W ∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"339 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75482004","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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