Commutative Algebra
New submissions
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New submissions for Tue, 14 May 24
- [1] arXiv:2405.06745 [pdf, ps, other]
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Title: Some Homological Conjectures Over Idealization RingsComments: 18 pages, 0 figuresSubjects: Commutative Algebra (math.AC)
Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes M$. First, if $N$ is a $R\ltimes k$-module, we show that the vanishing of $\operatorname{Ext}_{R\ltimes k}^{i}(N,N\oplus (R\ltimes k))$ for $i=1,2,3$ gives that $N$ is free, and this provides a sharpened version of the Auslander-Reiten conjecture over $R\ltimes k$. Also, we give a characterization of the Betti numbers of an $R$-module over the idealization
ring $R\ltimes M$ and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for $R\ltimes M$. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over $R$ implies the true over $R\ltimes M$. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when $R$ is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the
Zariski-Lipman conjecture holds for any ring $R$ provided the Betti numbers of the $R$-derivation module $\operatorname{Der}_k(R)$, seen as $R\ltimes k$-module, satisfy the inequality $\beta_{n}^{R\ltimes k}(\operatorname{Der}_k(R))\leq\beta_{n-1}^{R\ltimes k}(\operatorname{Der}_k(R))$ for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided. - [2] arXiv:2405.06781 [pdf, ps, other]
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Title: Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphsSubjects: Commutative Algebra (math.AC)
Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs. We also give formulas for the count of connected non-isomorphic spanning subgraphs of the complete bipartite graph K_{m,n} for which indm(G) = ordm(G) = 2, with an explicit expression for the count when m = 2,3,4 and m <= n.
- [3] arXiv:2405.06862 [pdf, ps, other]
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Title: On the factorial case of Huneke's conjecture for local cohomology modulesComments: 14 pagesSubjects: Commutative Algebra (math.AC)
A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$.
- [4] arXiv:2405.07000 [pdf, ps, other]
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Title: Multidegrees, families, and integral dependenceSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective degrees of rational maps are lower semicontinuous under specialization. We investigate various aspects of the polar multiplicities and Segre numbers of an ideal and introduce a new invariant that we call polar-Segre multiplicities. In terms of polar multiplicities and our new invariants, we provide a new integral dependence criterion for certain families of ideals. By giving specific examples, we show that the Segre numbers are the only invariants among the ones we consider that can detect integral dependence. Finally, we generalize the result of Gaffney and Gassler regarding the lexicographic upper semicontinuity of Segre numbers.
- [5] arXiv:2405.07002 [pdf, ps, other]
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Title: Computing the unit group of a commutative finite $\mathbb{Z}$-algebraComments: 16 pages; submitted to jGCCSubjects: Commutative Algebra (math.AC); Group Theory (math.GR); Rings and Algebras (math.RA)
For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully.
- [6] arXiv:2405.07271 [pdf, ps, other]
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Title: A note on $S$-coherent ringsAuthors: Xiaolei ZhangSubjects: Commutative Algebra (math.AC)
In this note, we show that a ring $R$ is $S$-coherent if and only if every finitely presented $R$-module is $S$-coherent, providing a positive answer to a question proposed in [D. Bennis, M. El Hajoui, {\it On $S$-coherence}, J. Korean Math. Soc. \textbf{55} (2018), no.6, 1499-1512].
- [7] arXiv:2405.07365 [pdf, ps, other]
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Title: Multidegrees of binomial edge idealsComments: 12 pages, 3 figures, submitted to the Proceedings of the AMSSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
Let $G$ be a simple graph with binomial edge ideal $J_G$. We prove how to calculate the multidegree of $J_G$ based on combinatorial properties of $G$. In particular, we study the set $S_{\min}(G)$ defined as the collection of subsets of vertices whose prime ideals have minimum codimension. We provide results which assist in determining $S_{\min}(G)$, then calculate $S_{\min}(G)$ for star, horned complete, barbell, cycle, wheel, and friendship graphs, and use the main result of the paper to obtain the multidegrees of their binomial edge ideals.
- [8] arXiv:2405.07566 [pdf, other]
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Title: Homological stability for general linear groups over Dedekind domainsAuthors: Oscar Randal-WilliamsComments: 18 pagesSubjects: Commutative Algebra (math.AC); Algebraic Topology (math.AT); K-Theory and Homology (math.KT)
We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces.
- [9] arXiv:2405.07989 [pdf, ps, other]
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Title: GPU-accelerated factorization sets in numerical semigroups via parallel bounded lexicographic streamsAuthors: Thomas BarronComments: 9 pages, 1 figureSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
We describe a method for parallelizing the lexicographic enumeration algorithm for the factorization set of an element in a numerical semigroup via bounds. This enables the use of GPU and distributed computing methods. We provide a CUDA implementation with measured runtimes.
Cross-lists for Tue, 14 May 24
- [10] arXiv:2405.07205 (cross-list from math.AG) [pdf, ps, other]
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Title: On Epimorphism and related problems for linear hypersurfacesSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two questions on linear polynomials of the form\\ $H:=\alpha(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether the affine variety $\mathbb{V}\in \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $\mathbb{V}$ is isomorphic to an affine space, then whether $H$ is a coordinate in $D$. In \cite{adv2}, the first two authors had addressed these questions when $\alpha$ is a monomial of the form $\alpha(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1,\, 1 \leqslant i \leqslant m$ and $F$ is of a certain type.
In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties.
In particular, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane (i.e., the affine variety $\mathbb{V}$ defined by $H$ is an affine space), then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. As a consequence we obtain a certain families of higher dimensional linear hyperplanes satisfying the Abhyankar-Sathaye conjecture on the Epimorphism Problem. Our results in arbitrary characteristic yield counter examples to the Zariski Cancellation Problem in positive characteristic. - [11] arXiv:2405.07939 (cross-list from math.AG) [pdf, ps, other]
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Title: Compact moduli of Calabi-Yau cones and Sasaki-Einstein spacesAuthors: Yuji OdakaComments: 56 pagesSubjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Commutative Algebra (math.AC); Differential Geometry (math.DG)
We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e., Sasaki-Einstein manifolds with singularities.
As a byproduct, it gives alternative algebraic construction of proper K-moduli of $\mathbb{Q}$-Fano varieties. In contrast to the previous algebraic proof of its properness ([BHLLX, LXZ]), we do not use the $\delta$-invariants ([FO, BJ]) nor the $L^2$-normalized Donaldson-Futaki invariants. We use the local normalized volume of [Li] and the higher $\Theta$-stable reduction instead.
Replacements for Tue, 14 May 24
- [12] arXiv:2302.07595 (replaced) [pdf, ps, other]
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Title: Macaulay's theorem for vector-spread algebrasComments: This is the final version of our paper, accepted for publication in the International Journal of Algebra and ComputationSubjects: Commutative Algebra (math.AC)
- [13] arXiv:2303.06953 (replaced) [pdf, ps, other]
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Title: Mapping cones of monomial ideals over exterior algebrasComments: This is the final version of our paper, published in Communications in AlgebraSubjects: Commutative Algebra (math.AC)
- [14] arXiv:2312.06124 (replaced) [pdf, ps, other]
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Title: On modules whose dual is of finite Gorenstein dimension: The $k$-torsion, homological criteria, and applications in modules of differentials of order $n$Comments: 28 pages. We added a few results and made some corrections. Comments and suggestions are welcomeSubjects: Commutative Algebra (math.AC)
- [15] arXiv:2405.01497 (replaced) [pdf, ps, other]
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Title: Auslander-Reiten conjecture for modules whose (self) dual has finite complete intersection dimensionComments: Theorem 3.6 in the first version is modified to Theorem 3.7 in this version. Comments and suggestions are welcome!Subjects: Commutative Algebra (math.AC)
- [16] arXiv:2306.07161 (replaced) [pdf, ps, other]
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Title: Minimal Terracini loci in projective spacesComments: 23 pages. Revised version. Title changedSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
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