{"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":"190 1","pages":""},"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":"48 1","pages":""},"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":"19 1","pages":""},"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":"9 1","pages":""},"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":"75 1","pages":""},"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}
For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of B. Ulrich, in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.
{"title":"Splitting the conormal module for licci ideals","authors":"Mark R. Johnson","doi":"10.1216/jca.2022.14.55","DOIUrl":"https://doi.org/10.1216/jca.2022.14.55","url":null,"abstract":"For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of B. Ulrich, in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"21 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75113873","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 we characterize the wellcovered property for: theta-ring and ring graphs. Furthermore, we prove that Cohen-Macaulayness, pure shellability and pure vertex decomposability are equivalent for theta-ring graphs. Also, we give a combinatorial characterization of these graphs.
{"title":"Well-covered and Cohen–Macaulay theta-ring graphs","authors":"Iván D. Castrillón, E. Reyes","doi":"10.1216/jca.2021.13.461","DOIUrl":"https://doi.org/10.1216/jca.2021.13.461","url":null,"abstract":"In this paper we characterize the wellcovered property for: theta-ring and ring graphs. Furthermore, we prove that Cohen-Macaulayness, pure shellability and pure vertex decomposability are equivalent for theta-ring graphs. Also, we give a combinatorial characterization of these graphs.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"18 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73298844","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 some classes of integral domains with finitely many star operations of finite type","authors":"Abdulilah Kadri, A. Mimouni","doi":"10.1216/jca.2021.13.489","DOIUrl":"https://doi.org/10.1216/jca.2021.13.489","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"27 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81450789","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}
This article is a natural continuation of our previous works [7] and [6]. In this article, we employ similar ideas as in [4] to provide an estimate of IC(K(X)|K, v) when (K(X)|K, v) is a valuation algebraic extension. Our central result is an analogue of [6, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.
{"title":"On the implicit constant fields and key polynomials for valuation algebraic extensions","authors":"Arpan Dutta","doi":"10.1216/jca.2022.14.515","DOIUrl":"https://doi.org/10.1216/jca.2022.14.515","url":null,"abstract":"This article is a natural continuation of our previous works [7] and [6]. In this article, we employ similar ideas as in [4] to provide an estimate of IC(K(X)|K, v) when (K(X)|K, v) is a valuation algebraic extension. Our central result is an analogue of [6, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"219 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75131837","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 work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules. The Veronese embedding injects the projective space Pn−1 into PN−1 by sending x = [x1 : x2 : · · · : xn] to the point with projective coordinates all possible monomials x1 1 . . . x in n of degree d, so N = ( n+d−1 d ) . The d-Veronese ring, S, is the coordinate ring of the image of the d-th Veronese embedding of Pn−1, with S = K[x1, . . . , xn]. The pinched Veronese map is another embedding of Pn−1, but this time the target space is PN−2 and the components of the image of x are all but one of the possible monomials. We denote such monomial by x. The coordinate ring of the latter image of Pn−1 is called pinched Veronese rings, Pn,d,m. The koszul property of the pinched Veronese rings was a trendy topic in literature. Peeva and Sturmfels asked whether the pinched Veronese ring P3,3,(1,1,1) is Koszul. A positive answer was given by Caviglia in [7], and then reproved by Caviglia and Conca in [8], and, after, in [9]; later, Tancer [21] generalized this result to Pn,n,(1,...,1). Vu used a combinatorial approach to prove that Pn,d,m is Koszul, unless d ≥ 3 and m is one of the permutations of (d− 2, 2, 0, . . . , 0), see [22]. 1991 AMS Mathematics subject classification. 13D02; 13D40; 05E99; 13C14, 13A02.
{"title":"Cohen–Macaulay property and linearity of pinched Veronese rings","authors":"Ornella Greco, Ivan Martino","doi":"10.1216/jca.2021.13.347","DOIUrl":"https://doi.org/10.1216/jca.2021.13.347","url":null,"abstract":"In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules. The Veronese embedding injects the projective space Pn−1 into PN−1 by sending x = [x1 : x2 : · · · : xn] to the point with projective coordinates all possible monomials x1 1 . . . x in n of degree d, so N = ( n+d−1 d ) . The d-Veronese ring, S, is the coordinate ring of the image of the d-th Veronese embedding of Pn−1, with S = K[x1, . . . , xn]. The pinched Veronese map is another embedding of Pn−1, but this time the target space is PN−2 and the components of the image of x are all but one of the possible monomials. We denote such monomial by x. The coordinate ring of the latter image of Pn−1 is called pinched Veronese rings, Pn,d,m. The koszul property of the pinched Veronese rings was a trendy topic in literature. Peeva and Sturmfels asked whether the pinched Veronese ring P3,3,(1,1,1) is Koszul. A positive answer was given by Caviglia in [7], and then reproved by Caviglia and Conca in [8], and, after, in [9]; later, Tancer [21] generalized this result to Pn,n,(1,...,1). Vu used a combinatorial approach to prove that Pn,d,m is Koszul, unless d ≥ 3 and m is one of the permutations of (d− 2, 2, 0, . . . , 0), see [22]. 1991 AMS Mathematics subject classification. 13D02; 13D40; 05E99; 13C14, 13A02.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"26 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90908579","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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