Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb P}^{s-1}$, over the field $\mathbb{F}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of $\mathbb{X}$. The main result of this paper is a formula for $I(\mathbb{X})$ that will allows us to compute: (i) the algebraic invariants of $I(\mathbb{X})$, and (ii) the basic parameters of the corresponding Reed-Muller-type code.
Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise disjoint even cycles. In this case, a fomula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph …
We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ ε-3}{2}(q-2),$$ …
The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest…
Given a finite simple graph one can associate the edge ideal. In this paper we prove that a graded Betti number of the edge ideal does not vanish if the original graph contains a set of complete bipartite subgraphs with some conditions. Also we give a combinatorial description for the projective dimension of the edge ideals of unmixed bipartite graphs.
We generalize some known results on the relation between the cohomological and projective dimension. Then we examine the set-theoretically Cohen-Macaulay ideals to find some cohomological characterization of these kind of ideals.
We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$
We discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we improve the computational algebra approach to the reverse engineering of gene regulatory networks.
Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as $ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.
Let K be a finite field. Let X* be a subset of the affine space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.
We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{α_1,...,α_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this …
Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number.
The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and its relationship to Buchsbaum rings are studied.
Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger--Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.
We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. The…
Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).
We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{ö}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weigh…
Let $X$ be a complete simplicial toric variety over a finite field $\mathbb{F}_q$ with homogeneous coordinate ring $S=\mathbb{F}_q[x_1,\dots,x_r]$ and split torus $T_X\cong (\mathbb{F}^*_q)^n$. We prove that vanishing ideal of a subset $Y$ of the torus $T_X$ is a lattice ideal if and only if $Y$ is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of subgroups of $T_X$ are r…
Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative mul…
All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of $n\times n$ rank $n-2$ matrices whose size $n-2$ principal minors vanish; this set has dimension $n^2-n-4$.
Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length …
Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of th…
We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The ob…
We show that under suitable hypothesis (which are sharp in certain sense) that the core of an m-primary ideal in a regular local ring of dimension d is equal to the adjoint (or multiplier) ideal of its d-th power, generalizing a result of Huneke and Swanson in dimension two. We also prove a version of this in the singular setting, which we show to be intimately related to the problem of finding global sections of ample line bundles on projective varieties. In particular, we show that a graded analog of our formula for core would imply a remarkable conjecture of Kawamata predicting that every a…
In this article, we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules. We apply our result to multi-Rees algebras of ideals. As a consequence, we obtain a result on normality of monomial ideals, which extends and improves the results of Reid-Roberts-Vitulli, Singla, and Sarkar-Verma.
In this paper, we focus on the initial degree and the vanishing of the Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of $J$. For edge ideals of graphs, a complete characterization of the vanishing of the Valabrega-Valla module is given. For higher degree ideals, we find classes which the Valabrega-Valla module vanishes. For the case that $J$ is the facet ideal of a clutter $\mathcal{C}$ and $I$ is the defining idea…
Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this paper we will describe several necessary and sufficient conditions such that the ideal $I$ becomes cohomologically complete intersections. In fact, as a technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all $i\neq c= \grade (I)$ is equivalent to the vanishing of the Betti numbers of $H^c_{I}(R)$. This gives a new characterization to check the cohomologically complete intersections property with the homological properties of the vanishing of Tor modules of $H^c_{I}(R)$.
In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as, Independence Theorem, Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing Theorem, among others. Besides, we present a generalized version of Melkersson Theorem about Artinianess of modules and a result concerning Artinianess of local cohomology modules.
Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We …
Let R be a Noetherian local ring with the maximal ideal m and dim R=1. In this paper, we shall prove that the module Ext^1_R(R/Q,R) does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m^2 kills the 0th local cohomology module H_m^0(R). The assertion is no longer true unless v(R) \leq 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.
Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that H^d_{\fa}(M)\simeq H^d_{\fM}(M)/\displaystyle{\sum_{n\in \mathbb{N}}}<\fM>(0:_{H^d_{\fM}(M)}\fa^n), where for an Artinian R-module A we put <\fM>A=\cap_{n\in \mathbb{N}} \fM^nA. As a consequence, it is proved that for all ideals \fa of R, there are only finitely many non-isomorphic top local cohomology modules H^d_{\fa}(M) having the same support. In addition, we est…
Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\fa}(X)=0$ if and only if $\dim \hat{R}/\fa \hat{R}+\fp>0$ for all prime ideals $\fp$ of $\hat{R}$ such that $\dim \hat{R}/\fp-\inf (X\otimes_R\hat{R})_{\fp}=d$.
Let $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal $\fm$, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $\{\HH{i}{\fm}{R/I^n}\}_{n\in \NN}$ for $i<\dim R/I$. We show that, when $\chara k= 0$, $R/I$ is Cohen-Macaulay, and $I$ is a complete intersection locally on $\Spec R \setminus\{\fm\}$, the lowest degrees of the modules $\{\HH{i}{\fm}{R/I^n}\}_{n\in \NN}$ are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $I/I^2$. Our resul…
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a…
We consider vanishing of Ext and Tor, especially over Artinian rings. In particular, we prove the Auslander-Reiten conjecture for all commutative local rings in which the cube of the maximal ideal is zero.
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ an $R$-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that $H^{\fa}_i(M)=0$ for all $i>\Mag_RM$. We prove this in several cases.
Let $(R,\fm)$ be a local ring and $I$ an ideal. The aim of the present paper is twofold. At first we continue the investigation to compare $\fgrade(I,R)$ with $\depth R/I$ and further we derive some results on the vanishing of Lyubeznik numbers.
We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.
We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of p…
Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain assumptions on the depth of associated graded ring $G(Q)$. For $2\leq i\leq d $, we show that (1) $e_i(Q)\leq 0$ provided depth $G(Q)\geq d-2$ and (2) $e_i(Q)\geq -λ_R(H_{\mathfrak{m}}^{d-1}(R))$ provided depth $G(Q)\geq d-1$. It is proved that $e_3(Q)\leq 0$. Further, we obtain a necessary condition for the vanishing of the last coefficient $e_d(Q)$. As a conse…
The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a polynomial with respect to a polynomial ideal I in the ring of multivariate polynomials over a field K, provided the field K is finite and the ideal I is a vanishing ideal. In order to use the concept of orthogonality, we introduce a symmetric bilinear form on a vector space over a finite field.
