{"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":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":"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}
. 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"Interpolation over ℤ and torsion in class groups","authors":"John D. Berman, D. Erman","doi":"10.1216/jca.2022.14.309","DOIUrl":"https://doi.org/10.1216/jca.2022.14.309","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86823081","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}
In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ we give a connection between complete sets of ABKPs for $w$ and MacLane-Vaqui'e chains of $w.$
{"title":"MAC LANE–VAQUIÉ CHAINS AND VALUATION-TRANSCENDENTAL EXTENSIONS","authors":"Sneha Mavi, Anuj Bishnoi","doi":"10.1216/jca.2023.15.249","DOIUrl":"https://doi.org/10.1216/jca.2023.15.249","url":null,"abstract":"In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ we give a connection between complete sets of ABKPs for $w$ and MacLane-Vaqui'e chains of $w.$","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87944685","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 establish a link between trace modules and rigidity in modules over Noetherian rings. We identify classes of modules which must have self-extensions and use the theory of trace ideals to verify the Auslander-Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.
{"title":"On the nonrigidity of trace modules","authors":"H. Lindo","doi":"10.1216/jca.2022.14.277","DOIUrl":"https://doi.org/10.1216/jca.2022.14.277","url":null,"abstract":"We establish a link between trace modules and rigidity in modules over Noetherian rings. We identify classes of modules which must have self-extensions and use the theory of trace ideals to verify the Auslander-Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74893224","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 G be a simple graph. In this article we show that if G is connected and R(I(G)) is normal, then reg(R(I(G))) ≤ α0(G), where α0(G) the vertex cover number of G. As a consequence, every normal König connected graph G, reg(R(I(G))) = mat(G), the matching number of G. For a gap-free graph G, we give various combinatorial upper bounds for reg(R(I(G))). As a consequence we give various sufficient conditions for the equality of reg(R(I(G))) and mat(G). Finally we show that if G is a chordal graph such that K[G] has q-linear resolution (q ≥ 4), then K[G] is a hypersurface, which proves the conjecture of Hibi-Matsuda-Tsuchiya [12, Conjecture 0.2], affirmatively for chordal graphs.
{"title":"On regularity bounds and linear resolutions of toric algebras of graphs","authors":"Rimpa Nandi, Ramakrishna Nanduri","doi":"10.1216/jca.2022.14.285","DOIUrl":"https://doi.org/10.1216/jca.2022.14.285","url":null,"abstract":"Let G be a simple graph. In this article we show that if G is connected and R(I(G)) is normal, then reg(R(I(G))) ≤ α0(G), where α0(G) the vertex cover number of G. As a consequence, every normal König connected graph G, reg(R(I(G))) = mat(G), the matching number of G. For a gap-free graph G, we give various combinatorial upper bounds for reg(R(I(G))). As a consequence we give various sufficient conditions for the equality of reg(R(I(G))) and mat(G). Finally we show that if G is a chordal graph such that K[G] has q-linear resolution (q ≥ 4), then K[G] is a hypersurface, which proves the conjecture of Hibi-Matsuda-Tsuchiya [12, Conjecture 0.2], affirmatively for chordal graphs.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90961580","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":"Module-theoretic characterizations of the ring of finite fractions of a commutative ring","authors":"Fangping Wang, D. Zhou, Dan Chen","doi":"10.1216/jca.2022.14.141","DOIUrl":"https://doi.org/10.1216/jca.2022.14.141","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86086199","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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