Questions tagged [monomial-ideals]

Use this tag for question involving monomial ideals in polynomial rings of several variables over a commutative ring. This tag should be used together with the tag of commutative algebra.

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On the definition of Chow rings for atomic lattices

In the following paper https://arxiv.org/pdf/math/0305142.pdf the authors introduce an algebra $D(\mathcal{L},\mathcal{G})$ (see Definition 3). However, they assert that "although D is defined ...
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Monomial in monomial ideal

Reading first time about monomial ideals and I am stuck with one of the very first results: Let $\mathbb{K} $ be a field and $\Lambda \subseteq \mathbb{Z}_{\ge 0}^n$. Given a monomial ideal $$  I= \...
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Proof that Monomial Ideals are finitely generated by monomials

I'm trying to prove that last part of Lemma 1.2.2 in Sturmfels' "Algorithms in Invariant Theory. For the induction step, we want to prove that n variate monomials M are finitely generated ...
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Minimal generating set of a monomial ideal of $k[x_1,\dots,x_n]$

Let $k$ be a field and consider the polynomial algebra $k[x_1,\dots,x_n]$. Suppose $I$ is a monomial ideal (generated by monomials). Since $k[x_1,\dots,x_n]$ is Noetherian we can choose a minimal ...
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Betti number of graded ideal and graded quotients

For a monomial ideal $I$ of a polynomial ring $S$ with degree $b$, Theorem 1.34 of the ``Combinatorial Commutative Algebra" says that $$\beta_{i,b}(I) = \beta_{i+1,b}(S/I)$$ with a proof stating ...
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How do i show that transitivity is required for a monomial order?

On Eisenbud - Commutative Algebra with a view Toward Algebraic geometry page 324 There is an example given why we must assume transitivity of a monomial order relation. this example, to me, seems ...
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Castelnuovo-Mumford Regularity of Sub-ideals $I \subset J$

Let $k$ be a field, let $I, J \subset k[x_1, \dots, x_n]$ be homogeneous ideals such that $I \subset J$, and let $\text{reg}(I), \text{reg}(J)$ be the Castelnuovo-Mumford regularity of $I, J$. We ...
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Hilbert series of $\mathbb Q[x_1,...,x_n]/I_\Delta+(x_1^2,...,x_n^2)$ , for Stanley-Reisner ideal $I_\Delta$

Let $\Delta$ be an abstract simplicial complex on vertex set $\{x_1,...,x_n\}$, fix the field $\mathbb Q$ and let $I_{\Delta}$ be the Stanley-Reisner ideal of $\mathbb Q[x_1,...,x_n]$ , and $\mathbb Q[...
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The intersection of squarefree monomial ideals is a squarefree monomial ideal

I want to show, that the intersection of two squarefree monomial ideals is again a squarefree monomial ideal. The definition of a squarefree monomial ideal I have is that the minimal set generating ...
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Compute Intersection of Two Ideals [duplicate]

In general, how do we compute the intersection of two monomial ideals? And could someone walk through an example in calculating the intersection of say $(x_1^2x_2, x_2x_4, x_3x_4x_5)\cap(x_1x_3^2, ...
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Minimal prime ideals and associated prime ideals

I am currently self-reading the book 'Monomial Ideals' by Herzog-Hibi. In lemma 1.3.5 of that book what is '$ P_j R_{p_i}$'? I know what $R_{p_i}$ is. It is the localization at $p_i$. But I am ...
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a corollary of dickson's lemma!

The question is if $J \subset I \subset R$ be ideals and we have that $\langle LT(I) \rangle = \langle LT(J) \rangle$ then $I = J$. I would like to show that for all monomials in I which are not in $\...
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Are minimal Groebner bases minimized bases?

With "minimal Groebner basis" I mean, fixed an ordering, a Groebner basis $G$ such that any proper subset of $G$ is no more a Groebner basis for the ideal $I(G)$ generated by $G$. With "...
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About the height of primes associated to a squarefree monomial ideal $I\subseteq J$

Edit Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ indeterminates with coefficients in a field $K$. For a monomial ideal $I$ of $S$, $G(I)$ denotes the minimal generating set of $I$. For ...
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Let $J=\langle x,y\rangle$ and $I=J^3=J\cdot J\cdot J$ in $\mathbb{Q}[x,y]$. (Two questions down below in the body...)

Let $J=\langle x,y\rangle$ and $I=J^3=J\cdot J\cdot J$ in $\mathbb{Q}[x,y]$. 1.1 The set of all monomials in $x, y$ is a basis of $\mathbb{Q}[x,y]$ as $\mathbb{Q}$-vector space. In particular, $x^2$ ...
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Proving that $\langle LT(I) \rangle$ is a monomial ideal where $I$ is an ideal.

I am trying to prove this statement and I am very new to monomial ideals. The definition of monomial ideal is stated like this in my book: An ideal $I ⊆ k[x_1,..., x_n]$ is a monomial ideal if there ...
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The ideal $(x^5,y^6,xy)$ cannot be generated by two elements [closed]

Is there any nice way to show an ideal $(x^5,y^6,xy)\subset F[x,y]$ where $F[x,y]$ is a polynomial ring with two variable over a field $F$ cannot be generated by two elements? I.e., there is no $f(x,y)...
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The Hilbert function and polynomial of $S = k[x_1, x_2, x_3, x_4]$ and $I = (x_1x_3, x_1x_4, x_2x_4)$ step clarification.

My professor based on pg. 320 - 321 of Eisenbud, wrote the following: Let $I = (m_1, \dots, m_l)$ be a minimal set of monomial generators, $I' = (m_1, \dots, m_{l-1}) \subsetneq I,$ and $d = \...
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Finding the syzygy (relation module) of a monomial ideal.

I have read pages 322-323 of "Commutative algebra, with a view toward algebraic geometry" by David Eisenbud, but it is still not much clear what are the steps of finding the syzygy. I am ...
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Lie groups in metric space

es $V_{x_0} \subset F$. I thought of saying that by taking the union of all these open sets given by the local diffeomorphism on every point, then I can say in a way I currently ignore that the ...
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Example of strict inclusion with respect to the product of initial ideals

I solved the following problem: Show that for any monomial order $>$ it is true that $\mathrm{in}_{>}(I)\mathrm{in}_{>}(J)\subseteq \mathrm{in}_{>}(IJ)$ for any two ideals $I,J$ of $k[x_{1}...
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On an Isomorphism of Semigroup Rings via Congruence Classes

Let $\mathbb Z_{\geq 0}$ denote the set of non-negative integers. Let $\mathbb Z_{\geq 0}^n$ denote the set of $n$-tuples of non-negative integers. (Theorem 2.1.5, Herzog, 1969) Given a finitely ...
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Regularity of generic initial ideals: The only associated prime of $I$ containing $x_r$ is the maximal ideal.

I was going through Eisenbud's proof of a result of Bayer and Stillman concerning the regularity of an ideal and its generic initial ideal, and I'm having trouble understanding his proof. The theorem ...
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Proof of two ideals are equal

How can we prove that the ideals $\,I=(x_1+x_2, x_2^2)\,$ and $\,J=(x_1+x_2, x_1^2)$ are equal? I was thinking of looking at $ {\rm Mon}(I) = {\rm Mon}(J)\,$. Thank you.
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Are $(x ^ 2, xy)$ and $(x) ∩ (x ^ 2, xy, y ^ 2)$ of $k [x, y]$ the same ideals in $k[x,y]$?

Let $k$ be a field. Are $(x ^ 2, xy)$ and $(x) ∩ (x ^ 2, xy, y ^ 2)$ of $k [x, y]$ the same ideals? The hint says $(x ^ 2, xy, y ^ 2)$ is the same as the subset of $k [x, y]$ of degree with 2 or ...
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Application of homogeneous version of Nakayama Lemma

I'm currently stuck in the proof of Proposition A.2.3. on page 267 of Monomial Ideals by Herzog and Hibi. First we consider the following commutative diagram $\require{AMScd}$ \begin{CD} F @>{\...
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On the definition of Power-product of elements of ideal

I am reading the paper Lefschetz Properties and Hyperplane arrangements by E. Palezzato and M. Torielli,(https://arxiv.org/pdf/1911.04083.pdf) and I am having problems finding the definition of power-...
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Monomials form a vector space basis for monomial ideal

Let $R=k[x_1,\ldots,x_n]$ for some field $k$. Let $I\subset R$ be a monomial ideal. That is, $I$ is generated by monomials in $R$. I want to show that the monomials in $I$ form a $k$-vector space ...
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Classification of monomials in a monomial ideal

We say a monomial ideal is one that is generated by monomials. Fix a field $k$ and consider the ring of polynomials $k[x_1,\ldots,x_n]$. Denote by $x^\alpha$ where $\alpha=(a_1,\ldots,a_n)\in\mathbb{Z}...
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How to prove this sufficient condition for when a monomial ideal is primary.

This answer does a good job at explaining that if $I$ is primary monomial ideal in $k[x_1, \dots, x_n]$, then $I = (x_{i_1}^{a_1}, \ldots, x_{i_m}^{a_m}, m_1, \ldots, m_k)$ where $m_1, \ldots, m_k$ ...
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Finding the length of the kernel of a map in Macaulay2

Given an equigenerated monomial ideal $I$ over a polynomial ring, I am trying to check if a sequence $L={l_1,\ldots,l_t}$ is almost $I$-regular, i.e. for each $i$, the kernel of the multiplication map ...
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On integral closedness of the multiplication of a monomial integrally closed ideal with the homogeneous maximal ideal

Consider the monomial ideal $I=(x^d, y^az^{d-a})$ in $\mathbb C [x,y,z]$ where $1\le a\le d-1$ are integers. Let $\mathfrak m=(x,y,z)$. Let $J=\overline I$ be the integral closure of $I$. If $J\...
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Betti numbers of edge ideal of disconnected graphs

Let $G$ be a finite simple graph and $I(G)$ be the edge ideal of $G$. Suppose we can write $G=H\sqcup K$ as a disjoint union of subgraphs. If $\beta_{i,j}$ is the graded betti numbers then is it true ...
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Is the Hilbert series of the monomial ideal $\langle x^3, xyz \rangle$ equal to $\frac{2x^3-x^5}{(1-x)^3}$?

Let $R$ be a polynomial ring of 3 variables over a field and take the monomial order lex. Let $I = \langle x^3, xyz \rangle$ There is just one syzygy $\left[\begin{matrix}yz \\-x^2\end{matrix}\right]...
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On an analogy of the highest generating degree and reduction of ideals

Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$ . Let $J \subseteq \mathfrak m$ be a homogenous ideal with $\sqrt J=\mathfrak m$ i.e. $\mathfrak m^n \subseteq J$ for some integer $n\ge 1$. Let $a\ge ...
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On a special property of powers of ideals motivated by Artin-Rees lemma

Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$. Let $I$ be a homogeneous ideal of $R$ with $\sqrt I=\mathfrak m$. Let $c\ge 1$ be the smallest integer such that $(I \cap \mathfrak m^c)\mathfrak m^{...
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A question about the common zeroes of a homogeneous polynomial and its partial derivatives

Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$ and $F \in K[X,Y,Z]$ a homogeneous polynomial of degree $d$. Let $F'_X, F'_Y, F'_Z$ denote the partial derivatives of $F$ and let $I=\...
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Entries of moment matrices are always nonnegative?

I am reading the following paper https://arxiv.org/pdf/1103.0486.pdf . Please see p.4, the part under Theorem 2.2. (just read from the bottom of p.3 to here). To my understanding, if the measure ...
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Gröbner Basis for a sum of ideals

Suppose that ideals $I$ and $J$ of $k[x_1,\dots,x_n]$ are given with $\{g_1,\dots,g_m\}$ and $\{f_1,\dots,f_n\}$ as their respective Grobner bases. Under what conditions is $\{g_1,\dots,g_m,f_1,\dots,...
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Is there any convenient way to index the monomials up to a given order by a number?

One canonical way of indexing a monomial of $n$ variables is to use a $n$-tuple of the powers, i.e., using $(\alpha_1, \ldots, \alpha_n)$ to index $x_1^{\alpha_1}\ldots x_n^{\alpha_n}$. I'm wondering ...
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Radical of $(xy,yz,xz)$ is $(xy,yz,xz)$ in the ring $k[x,y,z]$

I have to prove that in the ring $k[x,y,z]$ the radical of $(xy,yz,xz)$ is $(xy,yz,xz)$ itself. Can anyone give any pointers. I have sort of proved it by showing that $(xy,yz,xz)$ is the radical of $(...
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Kozsul Simplicial Complex a Cone

I'm really not understanding the idea of the (upper) Koszul simplicial complex: For a monomial ideal $I$ and a degree $\mathbf{b}\in\mathbb{N}^n$, define the upper simplicial Koszul complex as $$K^{\...
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Permutation of index and permutation of vector of powers of monomial

I am confused about the following easy stuff, Let $$\mathbf{x} =\begin{bmatrix} x_1 & \cdots & x_n\end{bmatrix}^T .$$ Suppose I have the following monomial $$\mathbf{x}^{\alpha}=x^{\alpha_1}...
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On the depths of symbolic powers of the Stanley-Reisner ideal of a bow-tie complex

Consider the polynomial ring $S=k[x_1,...,x_5]$. Consider the Stanley-Reisner ideal $I$ (i.e. the face ideal) of the simplicial complex which is a bow-tie $\Delta:=\left<x_1x_2x_3,x_3x_4x_5\right&...
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Primary decomposition of powers of a monomial (edge) ideal in a three variable polynomial ring

Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. As seen here Associated primes of the square of a monomial ideal , $Ass_R (R/J^n)=\{(x,y); (y,z);(z,x);(x,y,z)\}, \forall n \ge 2$. My ...
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Associated primes of the square of a monomial ideal

Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. How to show that $(x,y,z) \in \mathrm{Ass}_R (R/J^2)$ ?
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On the height of a square-free monomial ideal in a five variable polynomial ring

Consider the monomial ideal $I=(vw,wx,xy,yz,zv)$ in the polynomial ring $k[v,w,x,y,z]$. How do we find the height of the ideal $I$ ? Since $5=\dim k[v,w,x,y,z]=\mathrm{ht}(I)+ \dim (k[v,w,x,y,z]/I)$,...
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2 votes
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On a colon ideal in the polynomial ring $\mathbb R[x,y]$

Consider the ring $R=\mathbb R[x,y]$. Let $\mathfrak m=(x,y)$. Let $n\ge 3$ be an odd integer and let $I_n=(x^n,y^n)$. What is the smallest integer $s\ge 1 $ (obviously depending on $n$) such that $(...
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2 votes
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On monomial ideals and ring generated by monomials

Question 1: Is $(x^4,x^3y,x^2y^2,xy^3,y^4)$ a maximal ideal in $\mathbb C [ x^4,x^3y,x^2y^2,xy^3,y^4] $? Question 2: Are the ideals $(x^4,x^3y,x^2y^2,xy^3,y^4)$ and $(x^4,x^3y,xy^3,y^4)$ distinct in ...
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A question about initial ideals.

Let $R = k[x_1, \dots, x_m]$ be a polynomial ring over a field $k$ and $I, J$ be ideals of $R$. Further assume that $J$ is generated by the polynomials $f_1, \dots, f_r$. Fix a monomial order $<$ ...
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