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Homological dimensions of complexes over coherent regular rings 相干正则环上复合物的同调维数
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-12 DOI: arxiv-2409.08393
James Gillespie, Alina Iacob
{"title":"Homological dimensions of complexes over coherent regular rings","authors":"James Gillespie, Alina Iacob","doi":"arxiv-2409.08393","DOIUrl":"https://doi.org/arxiv-2409.08393","url":null,"abstract":"We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends\u0000from Noetherian to coherent rings. In particular, a coherent ring R is regular\u0000if and only if the injective (resp. projective) dimension of each complex X of\u0000R-modules agrees with its graded-injective (resp. graded-projective) dimension.\u0000The same is shown for the analogous dimensions based on FP-injective R-modules,\u0000and on flat R-modules.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249848","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
Monomial Cycles in Koszul Homology 科斯祖尔同构中的单项式循环
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-11 DOI: arxiv-2409.07583
Jacob Zoromski
{"title":"Monomial Cycles in Koszul Homology","authors":"Jacob Zoromski","doi":"arxiv-2409.07583","DOIUrl":"https://doi.org/arxiv-2409.07583","url":null,"abstract":"In this paper we study monomial cycles in Koszul homology over a monomial\u0000ring. The main result is that a monomial cycle is a boundary precisely when the\u0000monomial representing that cycle is contained in an ideal we introduce called\u0000the boundary ideal. As a consequence, we obtain necessary ideal-theoretic\u0000conditions for a monomial ideal to be Golod. We classify Golod monomial ideals\u0000in four variables in terms of these conditions. We further apply these\u0000conditions to symmetric monomial ideals, allowing us to classify Golod ideals\u0000generated by the permutations of one monomial. Lastly, we show that a class of\u0000ideals with linear quotients admit a basis for Koszul homology consisting of\u0000monomial cycles. This class includes the famous case of stable monomial ideals\u0000as well as new cases, such as symmetric shifted ideals.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185560","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
Tight closure of ideals on Witt rings 维特环上理想的严密封闭
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-10 DOI: arxiv-2409.06459
Shou Yoshikawa
{"title":"Tight closure of ideals on Witt rings","authors":"Shou Yoshikawa","doi":"arxiv-2409.06459","DOIUrl":"https://doi.org/arxiv-2409.06459","url":null,"abstract":"In this paper, we introduce the notions of tight closure of ideals on Witt\u0000rings and quasi-tightly closedness of system of parameters. By using the\u0000notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we\u0000study the relationship between the closure operator and integrally closure.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185565","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
Asymptotic depth of invariant chains of edge ideals 边理想不变链的渐近深度
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-10 DOI: arxiv-2409.06252
Tran Quang Hoa, Do Trong Hoang, Dinh Van Le, Hop D. Nguyen, Thai Thanh Nguyen
{"title":"Asymptotic depth of invariant chains of edge ideals","authors":"Tran Quang Hoa, Do Trong Hoang, Dinh Van Le, Hop D. Nguyen, Thai Thanh Nguyen","doi":"arxiv-2409.06252","DOIUrl":"https://doi.org/arxiv-2409.06252","url":null,"abstract":"We completely determine the asymptotic depth, equivalently, the asymptotic\u0000projective dimension of a chain of edge ideals that is invariant under the\u0000action of the monoid Inc of increasing functions on the positive integers. Our\u0000results and their proofs also reveal surprising combinatorial and topological\u0000properties of corresponding graphs and their independence complexes. In\u0000particular, we are able to determine the asymptotic behavior of all reduced\u0000homology groups of these independence complexes.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185561","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
Multiplicative Inequalities In Cluster Algebras Of Finite Type 有限类型簇代数中的乘法不等式
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-10 DOI: arxiv-2409.06642
Michael Gekhtman, Zachary Greenberg, Daniel Soskin
{"title":"Multiplicative Inequalities In Cluster Algebras Of Finite Type","authors":"Michael Gekhtman, Zachary Greenberg, Daniel Soskin","doi":"arxiv-2409.06642","DOIUrl":"https://doi.org/arxiv-2409.06642","url":null,"abstract":"Generalizing the notion of a multiplicative inequality among minors of a\u0000totally positive matrix, we describe, over full rank cluster algebras of finite\u0000type, the cone of Laurent monomials in cluster variables that are bounded as a\u0000real-valued function on the positive locus of the cluster variety. We prove\u0000that the extreme rays of this cone are the u-variables of the cluster algebra.\u0000Using this description, we prove that all bounded ratios are bounded by 1 and\u0000give a sufficient condition for all such ratios to be subtraction free. This\u0000allows us to show in Gr(2, n), Gr(3, 6), Gr(3, 7), Gr(3, 8) that every bounded\u0000Laurent monomial in Pl\"ucker coordinates factors into a positive integer\u0000combination of so-called primitive ratios. In Gr(4, 8) this factorization does\u0000not exists, but we provide the full list of extreme rays of the cone of bounded\u0000Laurent monomials in Pl\"ucker coordinates.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185566","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 computational approach to the study of finite-complement submonids of an affine cone 研究仿射锥的有限互补子单子的计算方法
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-10 DOI: arxiv-2409.06376
J. C. Rosales, R. Tapia-Ramos, A. Vigneron-Tenorio
{"title":"A computational approach to the study of finite-complement submonids of an affine cone","authors":"J. C. Rosales, R. Tapia-Ramos, A. Vigneron-Tenorio","doi":"arxiv-2409.06376","DOIUrl":"https://doi.org/arxiv-2409.06376","url":null,"abstract":"Let $mathcal{C}subseteq mathbb{N}^p$ be an integer cone. A\u0000$mathcal{C}$-semigroup $Ssubseteq mathcal{C}$ is an affine semigroup such\u0000that the set $mathcal{C}setminus S$ is finite. Such $mathcal{C}$-semigroups\u0000are central to our study. We develop new algorithms for computing\u0000$mathcal{C}$-semigroups with specified invariants, including genus, Frobenius\u0000element, and their combinations, among other invariants. To achieve this, we\u0000introduce a new class of $mathcal{C}$-semigroups, termed\u0000$mathcal{B}$-semigroups. By fixing the degree lexicographic order, we also\u0000research the embedding dimension for both ordinary and mult-embedded\u0000$mathbb{N}^2$-semigroups. These results are applied to test some\u0000generalizations of Wilf's conjecture.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185605","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 free approach to residual intersections 自由处理残差交叉点
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.05705
S. Hamid Hassanzadeh
{"title":"A free approach to residual intersections","authors":"S. Hamid Hassanzadeh","doi":"arxiv-2409.05705","DOIUrl":"https://doi.org/arxiv-2409.05705","url":null,"abstract":"This paper studies algebraic residual intersections in rings with Serre's\u0000condition ( S_{s} ). It demonstrates that residual intersections admit free\u0000approaches i.e. perfect subideal with the same radical. This fact leads to\u0000determining a uniform upper bound for the multiplicity of residual\u0000intersections. In positive characteristic, it follows that residual\u0000intersections are cohomologically complete intersection and, hence, their\u0000variety is connected in codimension one.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185564","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
Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles 科恩-麦考莱森林、周期和须周期的无平方幂
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.06021
Kanoy Kumar Das, Amit Roy, Kamalesh Saha
{"title":"Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles","authors":"Kanoy Kumar Das, Amit Roy, Kamalesh Saha","doi":"arxiv-2409.06021","DOIUrl":"https://doi.org/arxiv-2409.06021","url":null,"abstract":"Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal\u0000$I(G)$ of a graph $G$. In this article, we provide a precise formula for the\u0000depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show\u0000that for a Cohen-Macaualy forest $G$, the $k^{th}$ square-free power of $I(G)$\u0000is always Cohen-Macaulay, which is quite surprising since all ordinary powers\u0000of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges.\u0000Additionally, we provide tight bounds for the regularity and depth of\u0000$I(G)^{[k]}$ when $G$ is either a cycle or a whiskered cycle, which aids in\u0000identifying when such ideals have linear resolution. Furthermore, we provide\u0000combinatorial formulas for the depth of second square-free powers of edge\u0000ideals of cycles and whiskered cycles. We also obtained an explicit formula of\u0000the regularity of second square-free power for whiskered cycles.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185562","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
Classification and degenerations of small minimal border rank tensors via modules 通过模块实现小最小边界秩张量的分类和退化
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.06025
Jakub Jagiełła, Joachim Jelisiejew
{"title":"Classification and degenerations of small minimal border rank tensors via modules","authors":"Jakub Jagiełła, Joachim Jelisiejew","doi":"arxiv-2409.06025","DOIUrl":"https://doi.org/arxiv-2409.06025","url":null,"abstract":"We give a self-contained classification of $1_*$-generic minimal border rank\u0000tensors in $C^m otimes C^m otimes C^m$ for $m leq 5$. Together with previous\u0000results, this gives a classification of all minimal border rank tensors in $C^m\u0000otimes C^m otimes C^m$ for $m leq 5$: there are $37$ isomorphism classes. We\u0000fully describe possible degenerations among the tensors. We prove that there\u0000are no $1$-degenerate minimal border rank tensors in $C^m otimes C^m otimes\u0000C^m $ for $m leq 4$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185563","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
Stanley-Reisner Ideals with Pure Resolutions 斯坦利-赖斯纳理想与纯粹的决心
arXiv - MATH - Commutative Algebra Pub Date : 2024-09-09 DOI: arxiv-2409.05481
David Carey, Moty Katzman
{"title":"Stanley-Reisner Ideals with Pure Resolutions","authors":"David Carey, Moty Katzman","doi":"arxiv-2409.05481","DOIUrl":"https://doi.org/arxiv-2409.05481","url":null,"abstract":"We investigate Stanley-Reisner ideals with pure resolutions. To do this, we\u0000introduce the family of PR complexes, simplicial complexes whose dual\u0000Stanley-Reisner ideals have pure resolutions. We present two infinite families\u0000of highly-symmetric PR complexes. We also prove a partial analogue to the first\u0000Boij-S\"{o}derberg Conjecture for Stanley-Reisner ideals, by detailing an\u0000algorithm for constructing Stanley-Reisner ideals with pure Betti diagrams of\u0000any given shape, save for the initial shift $c_0$.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185586","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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