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":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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Let H be a Krull monoid with infinite cyclic class group G that we identify with Z. Let GP ⊆ G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H) ≤ min{− inf(GP ), sup(GP )}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H) > min{− inf(GP ), sup(GP )} − 3/2. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.
{"title":"Elasticities of Krull monoids with infinite cyclic class group","authors":"X. Zeng, Guixin Deng","doi":"10.1216/jca.2021.13.449","DOIUrl":"https://doi.org/10.1216/jca.2021.13.449","url":null,"abstract":"Let H be a Krull monoid with infinite cyclic class group G that we identify with Z. Let GP ⊆ G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H) ≤ min{− inf(GP ), sup(GP )}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H) > min{− inf(GP ), sup(GP )} − 3/2. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84219892","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 : 2021-07-30DOI: 10.1007/978-3-030-89694-2_4
W. Bruns, A. Conca, M. Varbaro
{"title":"Castelnuovo–Mumford Regularity and Powers","authors":"W. Bruns, A. Conca, M. Varbaro","doi":"10.1007/978-3-030-89694-2_4","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_4","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84044406","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 : 2021-07-29DOI: 10.1007/978-3-030-89694-2_6
Laurent Bus'e, M. Chardin
{"title":"Fibers of Rational Maps and Elimination Matrices: An Application Oriented Approach","authors":"Laurent Bus'e, M. Chardin","doi":"10.1007/978-3-030-89694-2_6","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_6","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80307886","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 : 2021-06-28DOI: 10.1007/978-3-030-89694-2_7
Lars Christensen, Oana Veliche, J. Weyman
{"title":"Three Takes on Almost Complete Intersection Ideals of Grade 3","authors":"Lars Christensen, Oana Veliche, J. Weyman","doi":"10.1007/978-3-030-89694-2_7","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_7","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88370279","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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