Ugo Bruzzo, Rodrigo Gondim, Rafael Holanda, William D. Montoya
{"title":"Cox-Gorenstein algebras","authors":"Ugo Bruzzo, Rodrigo Gondim, Rafael Holanda, William D. Montoya","doi":"arxiv-2407.17811","DOIUrl":"https://doi.org/arxiv-2407.17811","url":null,"abstract":"This paper is a first step in the study of nonstandard graded algebras having\u0000Poincar'e duality and their Lefschetz properties. We prove the equivalence\u0000between the toric setup and the G-graded one, generalize Macaulay-Matlis\u0000duality, introduce Lefschetz properties and prove a Hessian criteria in the\u0000G-graded setup. We prove a special case of the Codimension One Conjecture of\u0000Cattani-Cox-Dickenstein.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A characterization of monotypically supersymmetric polynomials","authors":"Grigory Chelnokov, Maxim Turevskii","doi":"arxiv-2407.18409","DOIUrl":"https://doi.org/arxiv-2407.18409","url":null,"abstract":"We introduce an object that has obvious similarity to the classical one - the\u0000algebra of supersymmetric polynomials. Despite the similarity, the known\u0000structure theorems on supersymmetric polynomials do not help in the study of\u0000the new object, so we prove their counterpart for the new object.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Numerical semigroups of coated odd elements","authors":"J. C. Rosales, M. B. Branco, M. A. Traesel","doi":"arxiv-2407.17153","DOIUrl":"https://doi.org/arxiv-2407.17153","url":null,"abstract":"A numerical semigroup $S$ is coated with odd elements (Coe-semigroup), if\u0000$left{x-1, x+1right}subseteq S$ for all odd element $x$ in $S$. In this\u0000note, we will study this kind of numerical semigroups. In particular, we are\u0000interested in the study of the Frobenius number, gender and embedding dimension\u0000of a numerical semigroup of this type.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semi-covariety of numerical semigroups","authors":"M. A. Moreno-Frías, J. C. Rosales","doi":"arxiv-2407.18984","DOIUrl":"https://doi.org/arxiv-2407.18984","url":null,"abstract":"The main aim of this work is to introduce and justify the study of\u0000semi-covarities. A {it semi-covariety} is a non-empty family $mathcal{F}$ of\u0000numerical semigroups such that it is closed under finite intersections, has a\u0000minimum, $min(mathcal{F}),$ and if $Sin mathcal{F}$ being $Sneq\u0000min(mathcal{F})$, then there is $xin S$ such that $Sbackslash {x}in\u0000mathcal{F}$. As examples, we will study the semi-covariety formed by all the\u0000numerical semigroups containing a fixed numerical semigroup, and the\u0000semi-covariety composed by all the numerical semigroups of coated odd elements\u0000and fixed Frobenius number.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The spectrum of a category of maximal Cohen-Macaulay modules","authors":"Naoya Hiramatsu","doi":"arxiv-2407.16913","DOIUrl":"https://doi.org/arxiv-2407.16913","url":null,"abstract":"We introduce an analog of the Ziegler spectrum for maximal Cohen-Macaulay\u0000modules over a complete Cohen-Macaulay local ring. We define a topology on the\u0000space of isomorphism classes of indecomposable maximal Cohen-Macaulay modules\u0000and investigate the topological structure. We also calculate the\u0000Cantor-Bendixson rank for a ring which is of CM_+-finite representation type.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"162 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Graded-Injective Modules and Bass Numbers of Veronese Submodules","authors":"Taylor Murray","doi":"arxiv-2407.17656","DOIUrl":"https://doi.org/arxiv-2407.17656","url":null,"abstract":"Let $R$ be a standard graded, finitely generated algebra over a field, and\u0000let $M$ be a graded module over $R$ with all Bass numbers finite. Set\u0000$(-)^{(n)}$ to be the $n$-th Veronese functor. We compute the Bass numbers of\u0000$M^{(n)}$ over the ring $R^{(n)}$ for all prime ideals of $R^{(n)}$ that are\u0000not the homogeneous maximal ideal in terms of the Bass numbers of $M$ over $R$.\u0000As an application to local cohomology modules, we determine the Bass numbers of\u0000$H_{Icap R^{(n)}}^i(R^{(n)})$ over the ring $R^{(n)}$ in the case where\u0000$H_I^i(R)$ has finite Bass numbers over $R$ and $I$ is a graded ideal.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Normal edge ideals of edge-weighted graphs","authors":"Thanh Vu, Guangjun Zhu","doi":"arxiv-2407.16118","DOIUrl":"https://doi.org/arxiv-2407.16118","url":null,"abstract":"We classify all normal edge ideals of edge-weighted graphs.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"350 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The reciprocal complement of a polynomial ring in several variables over a field","authors":"Neil Epstein, Lorenzo Guerrieri, K. Alan Loper","doi":"arxiv-2407.15637","DOIUrl":"https://doi.org/arxiv-2407.15637","url":null,"abstract":"The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring\u0000of its fraction field generated by the reciprocals of its nonzero elements.\u0000Many properties of $R(D)$ are determined when $D$ is a polynomial ring in\u0000$ngeq 2$ variables over a field. In particular, $R(D)$ is an $n$-dimensional,\u0000local, non-Noetherian, non-integrally closed, non-factorial, atomic G-domain,\u0000with infinitely many prime ideals at each height other than $0$ and $n$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rings for which general linear forms are exact zero divisors","authors":"Ayden Eddings, Adela Vraciu","doi":"arxiv-2407.16000","DOIUrl":"https://doi.org/arxiv-2407.16000","url":null,"abstract":"We investigate the standard graded $k$-algebras over a field $k$ of\u0000characteristic zero for which general linear forms are exact zero divisors. We\u0000formulate a conjecture regarding the Hilbert function of such rings. We prove\u0000our conjecture in the case when the ring is a quotient of a polynomial ring by\u0000a monomial idea, and also in the case when the ideal is generated in degree 2\u0000and all but one of the generators are monomials.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Humberto Muñoz-George, Enrique Reyes, Rafael H. Villarreal
{"title":"The v-numbers and linear presentations of ideals of covers of graphs","authors":"Humberto Muñoz-George, Enrique Reyes, Rafael H. Villarreal","doi":"arxiv-2407.15206","DOIUrl":"https://doi.org/arxiv-2407.15206","url":null,"abstract":"Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of\u0000this work are to study the {rm v}-number ${rm v}(J)$ of $J$ and to study when\u0000$J$ is linearly presented using combinatorics and commutative algebra. We\u0000classify when ${rm v}(J)$ attains its minimum and maximum possible values in\u0000terms of the vertex covers of the graph that satisfy the exchange property. If\u0000the cover ideal of a graph has a linear presentation, we express its v-number\u0000in terms of the covering number of the graph. We show necessary and sufficient\u0000conditions for the graph $mathcal{G}_J$ of $J$ to be connected. One of our\u0000main theorems shows that if $G$ has no induced 4-cycles, then $J$ is linearly\u0000presented. To prove this theorem, we show three results about unmixed K\"onig\u0000graphs related to some results already in the literature. For unmixed graphs\u0000without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly\u0000presented, and show that the columns of the linear syzygy matrix of $J$ are\u0000linearly independent if and only if $mathcal{G}_J$ has no strong $3$-cycles.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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