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Regularity of Koszul modules 科斯祖尔模块的正则性
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-18 DOI: arxiv-2409.11840
Tony J. Puthenpurakal
{"title":"Regularity of Koszul modules","authors":"Tony J. Puthenpurakal","doi":"arxiv-2409.11840","DOIUrl":"https://doi.org/arxiv-2409.11840","url":null,"abstract":"Let $K$ be a field and let $S = K[X_1, ldots, X_n]$. Let $I$ be a graded\u0000ideal in $S$ and let $M$ be a finitely generated graded $S$-module. We give\u0000upper bounds on the regularity of Koszul homology modules $H_i(I, M)$ for\u0000several classes of $I$ and $M$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249835","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}
引用次数: 0
The Existence of MacWilliams-Type Identities for the Lee, Homogeneous and Subfield Metric 李公设、同质公设和子场公设的麦克威廉斯类型同一性的存在性
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-18 DOI: arxiv-2409.11926
Jessica Bariffi, Giulia Cavicchioni, Violetta Weger
{"title":"The Existence of MacWilliams-Type Identities for the Lee, Homogeneous and Subfield Metric","authors":"Jessica Bariffi, Giulia Cavicchioni, Violetta Weger","doi":"arxiv-2409.11926","DOIUrl":"https://doi.org/arxiv-2409.11926","url":null,"abstract":"Famous results state that the classical MacWilliams identities fail for the\u0000Lee metric, the homogeneous metric and for the subfield metric, apart from some\u0000trivial cases. In this paper we change the classical idea of enumerating the\u0000codewords of the same weight and choose a finer way of partitioning the code\u0000that still contains all the information of the weight enumerator of the code.\u0000The considered decomposition allows for MacWilliams-type identities which hold\u0000for any additive weight over a finite chain ring. For the specific cases of the\u0000homogeneous and the subfield metric we then define a coarser partition for\u0000which the MacWilliams-type identities still hold. This result shows that one\u0000can, in fact, relate the code and the dual code in terms of their weights, even\u0000for these metrics. Finally, we derive Linear Programming bounds stemming from\u0000the MacWilliams-type identities presented.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"212 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268121","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}
引用次数: 0
Resolutions over strict complete resolutions 决议超过严格的完整决议
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-18 DOI: arxiv-2409.11877
Tony J. Puthenpurakal
{"title":"Resolutions over strict complete resolutions","authors":"Tony J. Puthenpurakal","doi":"arxiv-2409.11877","DOIUrl":"https://doi.org/arxiv-2409.11877","url":null,"abstract":"Let $(Q, mathfrak{n})$ be a regular local ring and let $f_1, ldots, f_c in\u0000mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, mathfrak{m}) =\u0000(Q/(mathbf{f}), mathfrak{n}/(mathbf{f}))$. Further assume that the initial\u0000forms $f_1^*, ldots, f_c^*$ form a $G(Q) = bigoplus_{n geq\u00000}mathfrak{n}^i/mathfrak{n}^{i+1}$-regular sequence. Without loss of any\u0000generality assume $ord_Q(f_1) geq ord_Q(f_2) geq cdots geq ord_Q(f_c)$. Let\u0000$M$ be a finitely generated $A$-module and let $(mathbb{F}, partial)$ be a\u0000minimal free resolution of $M$. Then we prove that $ord(partial_i) leq\u0000ord_Q(f_1) - 1$ for all $i gg 0$. We also construct an MCM $A$-module $M$ such\u0000that $ord(partial_{2i+1}) = ord_Q(f_1) - 1$ for all $i geq 0$. We also give a\u0000considerably simpler proof regarding the periodicity of ideals of minors of\u0000maps in a minimal free resolution of modules over arbitrary complete\u0000intersection rings (not necessarily strict).","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249815","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}
引用次数: 0
The complete integral closure of a Prüfer domain is a topological property 普吕弗域的完全积分闭包是一种拓扑性质
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-17 DOI: arxiv-2409.11189
Dario Spirito
{"title":"The complete integral closure of a Prüfer domain is a topological property","authors":"Dario Spirito","doi":"arxiv-2409.11189","DOIUrl":"https://doi.org/arxiv-2409.11189","url":null,"abstract":"We show that the prime spectrum of the complete integral closure $D^ast$ of\u0000a Pr\"ufer domain $D$ is completely determined by the Zariski topology on the\u0000spectrum $mathrm{Spec}(D)$ of $D$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249836","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}
引用次数: 0
Radical factorization in higher dimension 高维度辐射因式分解
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-16 DOI: arxiv-2409.10219
Dario Spirito
{"title":"Radical factorization in higher dimension","authors":"Dario Spirito","doi":"arxiv-2409.10219","DOIUrl":"https://doi.org/arxiv-2409.10219","url":null,"abstract":"We generalize the theory of radical factorization from almost Dedekind domain\u0000to strongly discrete Pr\"ufer domains; we show that, for a fixed subset $X$ of\u0000maximal ideals, the finitely generated ideals with $mathcal{V}(I)subseteq X$\u0000have radical factorization if and only if $X$ contains no critical maximal\u0000ideals with respect to $X$. We use these notions to prove that in the group\u0000$mathrm{Inv}(D)$ of the invertible ideals of a strongly discrete Pr\"ufer\u0000domains is often free: in particular, we show it when the spectrum of $D$ is\u0000Noetherian or when $D$ is a ring of integer-valued polynomials on a subset over\u0000a Dedekind domain.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249839","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}
引用次数: 0
Symbolic Powers and Symbolic Rees Algebras of Binomial Edge Ideals of Some Classes of Block Graphs 某些类块图的二项式边理想的符号幂和符号里斯代数
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-16 DOI: arxiv-2409.10137
Iman Jahani, Shamila Bayati, Farhad Rahmati
{"title":"Symbolic Powers and Symbolic Rees Algebras of Binomial Edge Ideals of Some Classes of Block Graphs","authors":"Iman Jahani, Shamila Bayati, Farhad Rahmati","doi":"arxiv-2409.10137","DOIUrl":"https://doi.org/arxiv-2409.10137","url":null,"abstract":"In this paper, we investigate some properties of symbolic powers and symbolic\u0000Rees algebras of binomial edge ideals associated with some classes of block\u0000graphs. First, it is shown that symbolic powers of binomial edge ideals of\u0000pendant cliques graphs coincide with the ordinary powers. Furthermore, we see\u0000that binomial edge ideals of a generalization of these graphs are symbolic\u0000$F$-split. Consequently, net-free generalized caterpillar graphs are also a\u0000class of block graphs with symbolic $F$-split binomial edge ideals. Finally, it\u0000turns out that symbolic Rees algebras of binomial edge ideals associated with\u0000these two classes, namely pendant cliques graphs and net-free generalized\u0000caterpillar graphs, are strongly $F$-regular.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249842","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}
引用次数: 0
Ideals, representations and a symmetrised Bernoulli triangle 理想、表征和对称伯努利三角形
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-16 DOI: arxiv-2409.10278
Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, Stavros Argyrios Papadakis
{"title":"Ideals, representations and a symmetrised Bernoulli triangle","authors":"Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, Stavros Argyrios Papadakis","doi":"arxiv-2409.10278","DOIUrl":"https://doi.org/arxiv-2409.10278","url":null,"abstract":"We study some representations of symmetric groups arising from a certain\u0000ideal in the coordinate ring of affine n-space. Our results give graded and\u0000representation-theoretic enhancements of sequence 337 of the Online\u0000Encyclopaedia of Integer Sequences, involving a symmetric version of the\u0000Bernoulli triangle.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249837","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}
引用次数: 0
Numerical characterizations for integral dependence of graded ideals 分级理想积分依赖性的数值特征
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-14 DOI: arxiv-2409.09346
Suprajo Das, Sudeshna Roy, Vijaylaxmi Trivedi
{"title":"Numerical characterizations for integral dependence of graded ideals","authors":"Suprajo Das, Sudeshna Roy, Vijaylaxmi Trivedi","doi":"arxiv-2409.09346","DOIUrl":"https://doi.org/arxiv-2409.09346","url":null,"abstract":"Let $R=oplus_{mgeq 0}R_m$ be a standard graded Noetherian domain over a\u0000field $R_0$ and $Isubseteq J$ be two graded ideals in $R$ such that\u0000$0<mbox{height};Ileq mbox{height};J <d$. Then we give a set of numerical\u0000characterizations of the integral dependence of $I$ and ${J}$ in terms of\u0000certain multiplicities. A novelty of the approach is that it does not involve\u0000localization and only requires checking computable and well-studied invariants. In particular, we show the following: let $S=R[Y]$, $mathsf{I} = IS$ and\u0000$mathsf{J} = JS$ and $bf d$ be the maximum generating degree of both $I$ and\u0000$J$. Then the following statements are equivalent. (1) $overline{I} = overline{J}$. (2) $varepsilon(I)=varepsilon(J)$ and $e_i(R[It]) = e_i(R[Jt])$ for all\u0000$0leq i <dim(R/I)$. (3) $ebig(R[It]_{Delta_{(c,1)}}big) = ebig(R[Jt]_{Delta_{(c,1)}}big)$\u0000and $ebig(S[mathsf{I}t]_{Delta_{(c,1)}}big) =\u0000ebig(S[mathsf{J}t]_{Delta_{(c,1)}}big)$ for some integer $c>{bf d}$. The statement $(2)$ generalizes the classical result of Rees. The statement\u0000$(3)$ gives the integral dependence criteria in terms of the Hilbert-Samuel\u0000multiplicities of certain standard graded domains over $R_0$. As a consequence\u0000of $(3)$, we also get an equivalent statement in terms of (Teissier) mixed\u0000multiplicities. Apart from several well-established results, the proofs of these results use\u0000the theory of density functions which was developed recently by the authors.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249846","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}
引用次数: 0
Demailly's Conjecture for general and very general points 一般点和非常一般点的德梅利猜想
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-13 DOI: arxiv-2409.08535
Sankhaneel Bisui, Dipendranath Mahato
{"title":"Demailly's Conjecture for general and very general points","authors":"Sankhaneel Bisui, Dipendranath Mahato","doi":"arxiv-2409.08535","DOIUrl":"https://doi.org/arxiv-2409.08535","url":null,"abstract":"We prove that at least $left( dfrac{(1+epsilon)2m}{N-1}+1+epsilon\u0000right)^N$, where $0leqslant epsilon <1$, many general points, satisfy\u0000Demailly's conjecture. Previously, it was known to be true for at least\u0000$(2m+2)^N$ many general points in arxiv.org/abs/2009.05022. We also study\u0000Demailly's conjecture for $m=3$ for ideal defining general and very general\u0000points.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268123","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}
引用次数: 0
A generalized depth formula for modules of finite quasi-projective dimension 有限准投影维数模块的广义深度公式
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-13 DOI: arxiv-2409.08996
Victor H. Jorge-Pérez, Paulo Martins, Victor D. Mendoza-Rubio
{"title":"A generalized depth formula for modules of finite quasi-projective dimension","authors":"Victor H. Jorge-Pérez, Paulo Martins, Victor D. Mendoza-Rubio","doi":"arxiv-2409.08996","DOIUrl":"https://doi.org/arxiv-2409.08996","url":null,"abstract":"In this paper, we present a generalized formulation of the depth formula for\u0000modules over Noetherian local rings, with an emphasis on quasi-projective\u0000dimension, extending the classical result of the depth formula originally\u0000demonstrated by Auslander, which involved projective dimension. Thus, we\u0000replace projective dimension with quasi-projective dimension and show that the\u0000general version of the depth formula remains valid under these conditions. This\u0000generalization of the depth formula allows us to obtain new consequences and\u0000applications.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249847","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}
引用次数: 0
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