We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends from Noetherian to coherent rings. In particular, a coherent ring R is regular if and only if the injective (resp. projective) dimension of each complex X of R-modules agrees with its graded-injective (resp. graded-projective) dimension. The same is shown for the analogous dimensions based on FP-injective R-modules, and on flat R-modules.
我们证明，伊阿科布-伊延格尔对阿夫罗莫夫-福克斯比问题的回答从诺特环扩展到了相干环。特别是，如果且只有当 R 模块的每个复数 X 的注入（或投影）维度与其分级注入（或分级投影）维度一致时，相干环 R 才是正则的。
{"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":null,"pages":null},"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}
In this paper we study monomial cycles in Koszul homology over a monomial ring. The main result is that a monomial cycle is a boundary precisely when the monomial representing that cycle is contained in an ideal we introduce called the boundary ideal. As a consequence, we obtain necessary ideal-theoretic conditions for a monomial ideal to be Golod. We classify Golod monomial ideals in four variables in terms of these conditions. We further apply these conditions to symmetric monomial ideals, allowing us to classify Golod ideals generated by the permutations of one monomial. Lastly, we show that a class of ideals with linear quotients admit a basis for Koszul homology consisting of monomial cycles. This class includes the famous case of stable monomial ideals as well as new cases, such as symmetric shifted ideals.
{"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":null,"pages":null},"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}
In this paper, we introduce the notions of tight closure of ideals on Witt rings and quasi-tightly closedness of system of parameters. By using the notions, we obtain a characterization of quasi-$F$-rationality. Furthermore, we study the relationship between the closure operator and integrally closure.
{"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":null,"pages":null},"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}
Tran Quang Hoa, Do Trong Hoang, Dinh Van Le, Hop D. Nguyen, Thai Thanh Nguyen
We completely determine the asymptotic depth, equivalently, the asymptotic projective dimension of a chain of edge ideals that is invariant under the action of the monoid Inc of increasing functions on the positive integers. Our results and their proofs also reveal surprising combinatorial and topological properties of corresponding graphs and their independence complexes. In particular, we are able to determine the asymptotic behavior of all reduced homology groups of these independence complexes.
{"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":null,"pages":null},"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}
Michael Gekhtman, Zachary Greenberg, Daniel Soskin
Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as a real-valued function on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u-variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr(2, n), Gr(3, 6), Gr(3, 7), Gr(3, 8) that every bounded Laurent monomial in Pl"ucker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr(4, 8) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Pl"ucker coordinates.
{"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":null,"pages":null},"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}
J. C. Rosales, R. Tapia-Ramos, A. Vigneron-Tenorio
Let $mathcal{C}subseteq mathbb{N}^p$ be an integer cone. A $mathcal{C}$-semigroup $Ssubseteq mathcal{C}$ is an affine semigroup such that the set $mathcal{C}setminus S$ is finite. Such $mathcal{C}$-semigroups are central to our study. We develop new algorithms for computing $mathcal{C}$-semigroups with specified invariants, including genus, Frobenius element, and their combinations, among other invariants. To achieve this, we introduce a new class of $mathcal{C}$-semigroups, termed $mathcal{B}$-semigroups. By fixing the degree lexicographic order, we also research the embedding dimension for both ordinary and mult-embedded $mathbb{N}^2$-semigroups. These results are applied to test some generalizations of Wilf's conjecture.
{"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":null,"pages":null},"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}
This paper studies algebraic residual intersections in rings with Serre's condition ( S_{s} ). It demonstrates that residual intersections admit free approaches i.e. perfect subideal with the same radical. This fact leads to determining a uniform upper bound for the multiplicity of residual intersections. In positive characteristic, it follows that residual intersections are cohomologically complete intersection and, hence, their variety is connected in codimension one.
{"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":null,"pages":null},"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}
Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaualy forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Additionally, we provide tight bounds for the regularity and depth of $I(G)^{[k]}$ when $G$ is either a cycle or a whiskered cycle, which aids in identifying when such ideals have linear resolution. Furthermore, we provide combinatorial formulas for the depth of second square-free powers of edge ideals of cycles and whiskered cycles. We also obtained an explicit formula of the regularity of second square-free power for whiskered cycles.
{"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":null,"pages":null},"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}
We give a self-contained classification of $1_*$-generic minimal border rank tensors in $C^m otimes C^m otimes C^m$ for $m leq 5$. Together with previous results, this gives a classification of all minimal border rank tensors in $C^m otimes C^m otimes C^m$ for $m leq 5$: there are $37$ isomorphism classes. We fully describe possible degenerations among the tensors. We prove that there are no $1$-degenerate minimal border rank tensors in $C^m otimes C^m otimes C^m $ for $m leq 4$.
{"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":null,"pages":null},"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}
We investigate Stanley-Reisner ideals with pure resolutions. To do this, we introduce the family of PR complexes, simplicial complexes whose dual Stanley-Reisner ideals have pure resolutions. We present two infinite families of highly-symmetric PR complexes. We also prove a partial analogue to the first Boij-S"{o}derberg Conjecture for Stanley-Reisner ideals, by detailing an algorithm for constructing Stanley-Reisner ideals with pure Betti diagrams of any given shape, save for the initial shift $c_0$.
{"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":null,"pages":null},"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}
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