Let $K$ be a field and let $S = K[X_1, ldots, X_n]$. Let $I$ be a graded ideal in $S$ and let $M$ be a finitely generated graded $S$-module. We give upper bounds on the regularity of Koszul homology modules $H_i(I, M)$ for several classes of $I$ and $M$.
{"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":null,"pages":null},"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}
Famous results state that the classical MacWilliams identities fail for the Lee metric, the homogeneous metric and for the subfield metric, apart from some trivial cases. In this paper we change the classical idea of enumerating the codewords of the same weight and choose a finer way of partitioning the code that still contains all the information of the weight enumerator of the code. The considered decomposition allows for MacWilliams-type identities which hold for any additive weight over a finite chain ring. For the specific cases of the homogeneous and the subfield metric we then define a coarser partition for which the MacWilliams-type identities still hold. This result shows that one can, in fact, relate the code and the dual code in terms of their weights, even for these metrics. Finally, we derive Linear Programming bounds stemming from the MacWilliams-type identities presented.
{"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":null,"pages":null},"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}
Let $(Q, mathfrak{n})$ be a regular local ring and let $f_1, ldots, f_c in mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, mathfrak{m}) = (Q/(mathbf{f}), mathfrak{n}/(mathbf{f}))$. Further assume that the initial forms $f_1^*, ldots, f_c^*$ form a $G(Q) = bigoplus_{n geq 0}mathfrak{n}^i/mathfrak{n}^{i+1}$-regular sequence. Without loss of any generality assume $ord_Q(f_1) geq ord_Q(f_2) geq cdots geq ord_Q(f_c)$. Let $M$ be a finitely generated $A$-module and let $(mathbb{F}, partial)$ be a minimal free resolution of $M$. Then we prove that $ord(partial_i) leq ord_Q(f_1) - 1$ for all $i gg 0$. We also construct an MCM $A$-module $M$ such that $ord(partial_{2i+1}) = ord_Q(f_1) - 1$ for all $i geq 0$. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).
{"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":null,"pages":null},"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}
We show that the prime spectrum of the complete integral closure $D^ast$ of a Pr"ufer domain $D$ is completely determined by the Zariski topology on the spectrum $mathrm{Spec}(D)$ of $D$.
{"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":null,"pages":null},"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}
We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Pr"ufer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $mathcal{V}(I)subseteq X$ have radical factorization if and only if $X$ contains no critical maximal ideals with respect to $X$. We use these notions to prove that in the group $mathrm{Inv}(D)$ of the invertible ideals of a strongly discrete Pr"ufer domains is often free: in particular, we show it when the spectrum of $D$ is Noetherian or when $D$ is a ring of integer-valued polynomials on a subset over a Dedekind domain.
{"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":null,"pages":null},"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}
In this paper, we investigate some properties of symbolic powers and symbolic Rees algebras of binomial edge ideals associated with some classes of block graphs. First, it is shown that symbolic powers of binomial edge ideals of pendant cliques graphs coincide with the ordinary powers. Furthermore, we see that binomial edge ideals of a generalization of these graphs are symbolic $F$-split. Consequently, net-free generalized caterpillar graphs are also a class of block graphs with symbolic $F$-split binomial edge ideals. Finally, it turns out that symbolic Rees algebras of binomial edge ideals associated with these two classes, namely pendant cliques graphs and net-free generalized caterpillar graphs, are strongly $F$-regular.
{"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":null,"pages":null},"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}
Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, Stavros Argyrios Papadakis
We study some representations of symmetric groups arising from a certain ideal in the coordinate ring of affine n-space. Our results give graded and representation-theoretic enhancements of sequence 337 of the Online Encyclopaedia of Integer Sequences, involving a symmetric version of the Bernoulli triangle.
我们研究了由仿射 n 空间坐标环中的某个理想产生的对称群的一些表示。我们的结果给出了《整数序列在线百科全书》中序列 337 的等级和表示论增强，涉及伯努利三角形的对称版本。
{"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":null,"pages":null},"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}
Let $R=oplus_{mgeq 0}R_m$ be a standard graded Noetherian domain over a field $R_0$ and $Isubseteq J$ be two graded ideals in $R$ such that $0{bf d}$. The statement $(2)$ generalizes the classical result of Rees. The statement $(3)$ gives the integral dependence criteria in terms of the Hilbert-Samuel multiplicities of certain standard graded domains over $R_0$. As a consequence of $(3)$, we also get an equivalent statement in terms of (Teissier) mixed multiplicities. Apart from several well-established results, the proofs of these results use the theory of density functions which was developed recently by the authors.
{"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":null,"pages":null},"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}
We prove that at least $left( dfrac{(1+epsilon)2m}{N-1}+1+epsilon right)^N$, where $0leqslant epsilon <1$, many general points, satisfy Demailly's conjecture. Previously, it was known to be true for at least $(2m+2)^N$ many general points in arxiv.org/abs/2009.05022. We also study Demailly's conjecture for $m=3$ for ideal defining general and very general points.
{"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":null,"pages":null},"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}
Victor H. Jorge-Pérez, Paulo Martins, Victor D. Mendoza-Rubio
In this paper, we present a generalized formulation of the depth formula for modules over Noetherian local rings, with an emphasis on quasi-projective dimension, extending the classical result of the depth formula originally demonstrated by Auslander, which involved projective dimension. Thus, we replace projective dimension with quasi-projective dimension and show that the general version of the depth formula remains valid under these conditions. This generalization of the depth formula allows us to obtain new consequences and applications.
{"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":null,"pages":null},"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}
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