Search

Consider ideals I of the form \[ I=(x_1^2,\dots, x_n^2)+\operatorname {RLex}(x_ix_j) \] where \operatorname {RLex}(x_ix_j) is the ideal generated by all the square-free monomials which are greater than or equal to x_ix_j in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of I. Using a theorem of Lindsey [Proc. Amer. Math. Soc. 139 (2011), no. 1, 79–92], this allows for a short proof that any algebra defined by I has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran [Colloq. Math. 173 (2023), no. 1, 1–8], this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property. [ABSTRACT FROM AUTHOR]

Suppose that K is a field, S = K [ x 1 , ... , x n ] or S = K [ [ x 1 , ... , x n ] ] and I is a monomial ideal of S. We study factorization properties of the ring R = S / I . In particular, we present conditions equivalent to R being présimplifiable, a bounded factorization ring, a finite factorization ring or a unique factorization ring. [ABSTRACT FROM AUTHOR]

This article explores a novel operation of a monomial ideal, termed duplication, which produces a monomial ideal I ⋄ from another I. This operation is inspired by and generalizes how vertex duplication affects the edge ideal of a graph. The main result describes a multigraded minimal free resolution of I ⋄ as long as we know a resolution for I. Consequently, we obtain formulas for the projective dimension and depth of I ⋄ in terms of the corresponding invariants of I. [ABSTRACT FROM AUTHOR]

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]

Let $ S $ be a polynomial ring over a field and $ I $ be an edge ideal associated with some classes of circulant graphs. We discussed the algebraic invariants, namely, regularity, projective dimension, depth, and the Stanley depth of $ S/I. $

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen–Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen–Macaulay affine semigroup rings. [ABSTRACT FROM AUTHOR]

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A (Δ) = k [ x 1 , ... , x n ] / 〈 x 1 2 , ... , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A (Δ) in terms of the simplicial complex Δ. We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A (Δ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization. [ABSTRACT FROM AUTHOR]

Let S be a polynomial ring over a field and I be an edge ideal associated with some classes of circulant graphs. We discussed the algebraic invariants, namely, regularity, projective dimension, depth, and the Stanley depth of S=I. [ABSTRACT FROM AUTHOR]

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

Let k be a field, let A and B be polynomial rings over k , and let S = A ⊗ k B . Let I ⊆ A and J ⊆ B be monomial ideals. We establish a binomial expansion for rational powers of I + J ⊆ S in terms of those of I and J. Particularly, for u ∈ Q + , we prove that (I + J) u = ∑ 0 ≤ ω ≤ u , ω ∈ Q I ω J u - ω , and that the sum on the right-hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of I and J. We further give sufficient conditions for this formula to hold for the integral closures of powers of I + J in terms of those of I and J. Under these conditions, we provide explicit formulas for the depth and regularity of (I + J) k ¯ in terms of those of powers of I and J. [ABSTRACT FROM AUTHOR]

Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I. [ABSTRACT FROM AUTHOR]

Let G be a simple finite graph. A famous theorem of Dirac says that G is chordal if and only if G admits a perfect elimination order. It is known by Fröberg that the edge ideal I(G) of G has a linear resolution if and only if the complementary graph G c of G is chordal. In this article, we discuss some algebraic consequences of Dirac's theorem in the theory of homological shift ideals of edge ideals. Recall that if I is a monomial ideal, H S k (I) is the monomial ideal generated by the kth multigraded shifts of I. We prove that H S 1 (I) has linear quotients, for any monomial ideal I with linear quotients generated in a single degree. For and edge ideal I(G) with linear quotients, it is not true that H S k (I (G)) has linear quotients for all k ≥ 0 . On the other hand, if G c is a proper interval graph or a forest, we prove that this is the case. Finally, we discuss a conjecture of Bandari, Bayati, and Herzog that predicts that if I is polymatroidal, H S k (I) is polymatroidal too, for all k ≥ 0 . We are able to prove that this conjecture holds for all polymatroidal ideals generated in degree two. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is the comparison between an ideal and its integral closure and how to obtain the integral closure of ideals in different ways. We know that integral closure of a monomial ideal is again a monomial ideal and monomial ideals are associated to graphs. As graphs are widely used in computer science and cryptography in which there are only binary numbers, we introduce the associated matrix to a graph only with digits of 0 and 1. Moreover, we compare and compute the index of stability of associated prime ideals with index of stability of the integral closure of associated prime ideals. [ABSTRACT FROM AUTHOR]

Let S be a ring of polynomials in finitely many variables over a field. In this paper, we give lower bounds for depth and Stanley depth of modules of the type S/It for t = 1, where I is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered. [ABSTRACT FROM AUTHOR]

In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered.

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals J ⊆ I in a polynomial ring over a field . We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of . Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of generic monomial ideals generated by the monomial parametrization of rational surfaces. First, we describe the necessary and sufficient conditions for this family of monomial ideals to be generic. Then, we classify all the possible minimal graded free resolutions of this family. [ABSTRACT FROM AUTHOR]

We give a new method to construct minimal free resolutions of all monomial ideals. This method relies on two concepts: one is the well-known lcm-lattice of a monomial ideal; the other is a new concept called Taylor basis, which describes how a minimal free resolution can be embedded in Taylor resolution. An approximation formula for minimal free resolutions of all monomial ideals is also obtained. [ABSTRACT FROM AUTHOR]

We introduce pretty k-clean monomial ideals and k-decomposable multicomplexes, respectively, as the extensions of the notions of k-clean monomial ideals and k-decomposable simplicial complexes. We show that a multicomplex Γ is k-decomposable if and only if its associated monomial ideal I(Γ) is pretty k-clean. Also, we prove that an arbitrary monomial ideal I is pretty k-clean if and only if its polarization IP is k-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the first author. [ABSTRACT FROM AUTHOR]

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δI of I and give bounds for its stabilization point, rI, when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

In this paper we study the tangent space to the Hilbert scheme \mathrm {Hilb}^d \mathbf {P}^3, motivated by Haiman's work on \mathrm {Hilb}^d \mathbf {P}^2 and by a long-standing conjecture of Briançon and Iarrobino [J. Algebra 55 (1978), pp. 536–544] on the most singular point in \mathrm {Hilb}^d \mathbf {P}^n. For points parametrizing monomial subschemes, we consider a decomposition of the tangent space into six distinguished subspaces, and show that a fat point exhibits an extremal behavior in this respect. This decomposition is also used to characterize smooth monomial points on the Hilbert scheme. We prove the first Briançon-Iarrobino conjecture up to a factor of \frac {4}{3}, and improve the known asymptotic bound on the dimension of \mathrm {Hilb}^d \mathbf {P}^3. Furthermore, we construct infinitely many counterexamples to the second Briançon-Iarrobino conjecture, and we also settle a weaker conjecture of Sturmfels in the negative. [ABSTRACT FROM AUTHOR]

In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered. [ABSTRACT FROM AUTHOR]

We give effective lower and upper bounds on the maximal generating degree d (I n ¯) and the Castelnuovo–Mumford regularity reg (I n ¯) of the integral closure of a power I n of a monomial ideal I for all n and determine the leading coefficient of the asymptotically linear functions d (I n ¯) and reg (I n ¯) . A number n 0 is also given such that d (I n ¯) becomes a linear function of n when n ≥ n 0 . [ABSTRACT FROM AUTHOR]

Let I be a two-dimensional squarefree monomial ideal in a polynomial ring R = K [ x 1 , ... , x n ] , where K is a field. In this paper, we give explicit formulas for the extremal Betti numbers of the m th symbolic power of I for all m ≥ 1. As a consequence, we characterize the rings R / I (m) which are pseudo-Gorenstein as sense of Ene et al. [Pseudo-Gorenstein and level Hibi rings, J. Algebra431 (2015) 138–161]. We also provide a complete classification for the level property of the second symbolic power I (2) . In particular, we obtain a new algebraic-property of the unknown Moore graph of degree 57. [ABSTRACT FROM AUTHOR]

Let (R , m) be an equicharacteristic regular local ring of dimension d ≥ 3 and m = (x 1 , x 2 , ... , x d) R , and consider the sequence of d-dimensional local monomial quadratic transformations R : = R 0 ⊂ R 1 ⊂ R 2 ⊂ ⋯ ⊂ R n − 1 ⊂ R n (n ≥ 2) , where R i + 1 is a local monomial quadratic transformation of Ri for i = 0 , 1 , ... , n − 1. We prove that Rees R 0 (∏ i ≥ 0 n P R 0 R i k i ) = ∪ i ≥ 0 n ( Rees R 0 (P R 0 R i k i )) , where k i ≥ 0 and P R 0 R i is the special * -simple complete ideal associated with R 0 ≺ R i. Let I and J be finitely supported complete m -primary monomial ideals in R. We show that if the set B P (I J) of base points of the ideal IJ is linearly ordered, then Rees R (I J) = Rees R (I) ∪ Rees R (J). [ABSTRACT FROM AUTHOR]

compute depth and Stanley depth of the quotient rings of the edge ideals associated with different classes of graphs. These classes include some lobster trees and unicyclic graphs. We show that the values of depth and Stanley depth are equal for the classes we considered. [ABSTRACT FROM AUTHOR]

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S = k[xi,..., xn] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then pd(S/M) ≤ n -- i. [ABSTRACT FROM AUTHOR]

Let I be a square-free monomial ideal of projective dimension one. Starting with the Taylor complex on the generators of I r , we use discrete Morse theory to describe a CW complex that supports a minimal free resolution of I r . To do so, we concretely describe the acyclic matching on the faces of the Taylor complex. [ABSTRACT FROM AUTHOR]

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer N ∗ is determined such that the optima of these integer programs are a quasi-linear function of n for all n ≥ N ∗ . Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo–Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal. [ABSTRACT FROM AUTHOR]

Let S = K [ x 1 , ... , x n ] , where K is a field, and t i denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove: If I is a monomial ideal of S and a ≥ b - 1 ≥ 0 are integers such that a + b ≤ proj dim (S / I) , then t a + b ≤ t a + t 1 + t 2 + ⋯ + t b - b (b - 1) 2. If I = I Δ where Δ is a simplicial complex such that dim (Δ) < t a - a or dim (Δ) < t b - b , then t a + b ≤ t a + t b. If I is a monomial ideal that minimally generated by m 1 , ... , m r such that lcm (m 1 , ... , m r) lcm (m 1 , ... , m ^ i , ... , m r) ∉ K for all i, where m ^ i means that m i is omitted, then t a + b ≤ t a + t b for all a , b ≥ 0 with a + b ≤ proj dim (S / I) . [ABSTRACT FROM AUTHOR]

In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph G. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal I(G). We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton's polyhedron is the irreducible polyhedron is presented using integral closure. [ABSTRACT FROM AUTHOR]

Let R = K [ x 1 , … , x n ] be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog (Eur. J. Combin.34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) height (I) = n - 1 ; (ii) I contains at least n - 3 pure powers of the variables x 1 d , … , x n - 3 d ; (iii) I is a monomial ideal in at most four variables. [ABSTRACT FROM AUTHOR]

for each squarefree monomial ideal I ⊂ S = k[x1, ..., xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on GI, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where G Δ ≅ G I Δ ∨ is a cycle or a tree. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ ν , h ] -ary trees and some unicyclic graphs., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph. [ABSTRACT FROM AUTHOR]

Let S = K [ x 1 , ⋯ , x m , y 1 , ⋯ , y n ] be the standard bigraded polynomial ring over a field K. Let M be a finitely generated bigraded S-module and Q = (y 1 , ⋯ , y n) . We say M has maximal depth with respect to Q if there is an associated prime p of M such that grade (Q , M) = cd (Q , S / p) . In this paper, we study finitely generated bigraded modules with maximal depth with respect to Q. It is shown that sequentially Cohen–Macaulay modules with respect to Q have maximal depth with respect to Q. In fact, maximal depth property generalizes the concept of sequentially Cohen–Macaulayness. Next, we show that if M has maximal depth with respect to Q with grade (Q , M) > 0 , then H Q grade (Q , M) (M) is not finitely generated. As a consequence, "generalized Cohen–Macaulay modules with respect to Q" having "maximal depth with respect to Q" are Cohen–Macaulay with respect to Q. All hypersurface rings that have maximal depth with respect to Q are classified. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of monomial ideals with respect to a regular sequence on a commutative Noetherian ring R. Let x := x1 ... xd be a regular R-sequence contained in the Jacobson radical of R. We first show that each monomial ideal a of R with respect to x has a unique decomposition as an irredundant finite intersection of ideals of the ... are positive integers. As a consequence, it follows that a is an irreducible ideal if and only if it is a generalized-parameter ideal. In addition, it is shown that if xR is a prime ideal, then the radical of a and the symbolic powers of a are monomials. Finally, we prove that if a is a square-free monomial ideal such that ... for all integers k > 1, then a is normal and a(k) = ak, where a(k) denotes the kth symbolic power of a. [ABSTRACT FROM AUTHOR]

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound μ (I 2) ≥ 9 for the number of minimal generators of I 2 with μ (I) ≥ 6 . Recently, Gasanova constructed monomial ideals such that μ (I) > μ (I n) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that μ (I) > μ (I 2) > ⋯ > μ (I n) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function μ (I k) . The monomial ideals also give a peculiar example such that the Cohen–Macaulay type (or the index of irreducibility) of R / I n descends. [ABSTRACT FROM AUTHOR]

School of Natural Sciences, National University of Sciences and Technology Islamabad, Sector H-12, Islamabad, Pakistan