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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17 views

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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1answer
22 views

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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1answer
58 views

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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1answer
37 views

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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1answer
55 views

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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138 views

On a colon ideal in the polynomial ring $\mathbb R[x,y]$ [closed]

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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50 views

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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43 views

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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46 views

Quotient of monomial ideals

I've done a lot of calculations but they all seem to lead nowhere: Consider the ring of multivariate polynomials with field coefficients $K[X_1,\dots,X_n]$, and two monomial ideals, say $\alpha=\...
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52 views

Complete Intersection ideal for which $\mbox{in}_<(I)$ is not even Cohen-Macaulay

I would like to find a example of a graded ideal $I\subset k[x_1,\cdots,x_n]$ for which $\mbox{in}_<(I)$ is not even Cohen-Macaulay (for some monomial order). I have tried to find such a ideal ...
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59 views

Complete Intersection ideal whose initial ideal is not even Cohen-Macaulay

I am trying to find an example, such a example is asked in exercise 3.3 of the book Monomial Ideals (Herzog & Hibi). I would like to find a graded ideal $I\subset k[x_1,\cdots,x_n]$ which is ...
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1answer
31 views

Assuming that $I,J\subset S$ are monomial ideals, is it true that $I+J$ is monomial? What about $IJ$ and $I\cap J$?

Let $\mathbb{k}$ be a field and $S=\mathbb{k}[x_1,\dots,x_k]$ be a ring of polynomials over $\mathbb{k}$. Assuming that ideals $I,J\subset S$ are monomial, is it true that $I+J$ is monomial? What ...
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1answer
26 views

Showing $I:\langle v_j \rangle \subseteq \langle \{ u_i / \gcd(u_i,v_j) \ : \ i=1,\dots, r \} \rangle$, where $I=\langle u_1, \dots, u_r \rangle$

Consider a field $K$, a monomial $v_j \in K[x_1, \dots, x_n]$, and a monomial ideal $I=\langle u_1, u_2\dots, u_r \rangle$. I'm having trouble showing $I:\langle v_j \rangle \subseteq \langle \{ u_i / ...
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53 views

Reduces to $0$ of $S$-polynomial.

It is problem 2.17 in the book Gröbner Bases in Commutative Algebra, by Ene and Herzog. Let $f,g \in S$ such that $\textrm{in}_{<}(f)$ and $\textrm{in}_{<}(g)$ are relatively prime and let $u$...
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1answer
30 views

Monomial ideal which is not saturated.

The problem is below. Let $I$ be a monomial ideal and let $J= (x_{1}, \dots, x_{r}).$ Show that $I:J \neq I$ if there exist integers $a_{i} >0$ such that $x_{i}^{a_{i}} \in G(I)$ for $i=1, \dots,...
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1answer
50 views

Let $I = (X,Y) \subset k[X,Y]$ then $\dim_k(k[X,Y]/I^n) = \frac{n(n+1)}{2}$ [closed]

Why we have this: Let $I = (X,Y) \subset k[X,Y]$ then $\dim_k(k[X,Y]/I^n) = 1+2+3+...n = \frac{n(n+1)}{2}$ This is no clear for me.
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If the initial monomial ideal $in_<(I)$ is square free, then $I$ is a radical ideal.

I have to prove the following statement: Let $I$ be an ideal and < be any monomial order. If the initial monomial ideal $in_<(I)$ is square free, then $I$ is a radical ideal. If I can prove ...
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1answer
30 views

Prove $\langle x \rangle + \cap P_i = \cap (\langle x \rangle + P_i)$ for monomials ideals.

I'm trying to follow the proof of this fact on this book on Gröbner basis. At one point of the proof they use this equality that I do not fully understad. I wonder if it is evident: Given prime ...
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163 views

Finite set of generators of monomial ideal form Gröbner basis

Given a set of monomials $\{G_1,\ldots,G_t\}$ generating a non-null monomial ideal $I \le K[X_1,\ldots,K_n]$ I would like to check that they form a Gröbner basis. This is done by Buchberger's ...
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86 views

Monomial ideal; Dickson's lemma

Let $ I \neq {\{0}\}$ be a monomial ideal of $\mathbb{Q}[X_1, ..., X_n]$. Show that $I$ contains a monomial $m$ such that $m$ is divisible by exactly $2015$ other monomials contained in $I$. Now, at ...
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2answers
68 views

Prove that $\mathbf{x^b}\in \langle\mathbf{x^{a_1},…,x^{a_k}} \rangle\iff \exists j\in \{1,…,k\}:\mathbf{x^{a_j}\mid x^b} $

We want to prove the following lemma: Lemma. Let $K$ be a field and $I:= \langle\mathbf{x^{a_1},...,x^{a_k}} \rangle$ be an ideal of the polynomial ring $K[x_1,...,x_n]$ (which is generated by ...
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0answers
157 views

Finding the minimal primary decomposition of a monomial ideal

I have a question regarding, how to find/calculate the minimal primary decomposition given an ideal $\mathfrak{a}$. In my case, let $k$ be a field. Consider the polynomial ring $A:= k[X,Y,Z]$ and the ...
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1answer
135 views

Hilbert scheme of $n$ point action, torus action fixed points

I am trying to understand the torus action on the Hilbert Scheme. If we have an ideal in the polynomial ring $\mathbb{C}[x,y]$, then there is an action of $(\mathbb{C}^*)^2$ on I defined as $(t_1,t_2)...
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2answers
113 views

Non-monomial ideal with monomial radical

I'm given the ideal $I = \langle x+y,x^2y^2 \rangle$ as an example of an ideal whose radical $\sqrt{I} = \langle x,y \rangle$ is monomial even $I $ is not monomial itself. I'm trying to fill in the ...
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1answer
196 views

Intersection distributes over sum for monomial ideals.

I want to show that for monomials ideals the intersection distributes over the sum in a basic expression like this: $$I \cap (J+K) = (I \cap J) + (I \cap K).$$ How can I prove this?
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2answers
67 views

Is $\langle X^2Y + Y^2X \rangle$ a monomial ideal in $K[X,Y]$?

Is $\langle X^2Y + Y^2X \rangle$ a monomial ideal in $K[X,Y]$ (where $K$ is a field)? I have a feeling the answer is no, but I am having trouble justifying it. Can anyone show me how to prove this? ...
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0answers
50 views

Are two monomial ideals equal iff they contain the same elements of minimal degree?

In working with two ideals in $k[x_1,\ldots,x_n]$ where $k$ is a field I know that the all elements in both have degree greater than or equal to $x$ and that every element of degree $x$ in ideal $A$ ...
3
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1answer
114 views

monomial ideals

I was reading the book $\textit{Ideals, Varieties, and Algorithms}$ by Cox, Little, and O'Shea, and on Chapter 2 page 71 the have the following lemma Lemma 3: Let $I$ be a monomial ideal, and let $f \...
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1answer
433 views

Property of colon ideal: $I: (J+K)=(I:J)\cap(I:K)$

Let $I,J,K$ are ideals in a commutative ring with unity $R$ Then $I:(J+K)=(I:J)\cap(I:K)$ My solution is if $a\in I: (J+K)$ then $a(j+k)\in I$. For all $j+k\in J+K$ $aj+ak\in I$ but $aj \in I$ and $...
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1answer
33 views

Definition of zero in monomial ideals

I am studying monomial ideals and I have a problem with this definition: Let R=A$[X_1.....X_k]$, and: $$0=(\phi)R $$ How this ideal contains 0? Thanks
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56 views

Asymptotic generator degree of monomial ideals

Let $I$ be a homogeneous ideal in the polynomial ring $K[x_1,\cdots, x_n]$. The asymptotic generator degree of $I$ is defined to be the minimal number $d$ such that $I$ is integral over $I_{\leq d}$. ...
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1answer
61 views

Monomial ideals Lemma

There's a Lemma about monomial ideals that says: "Let $I=\left<x^α \mid \ α ∈ A\right>$ be a monomial ideal. Then a monomial $x^β$ lies in $I$ if and only if $x^β$ is divisible by $x^α$ for ...
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1answer
183 views

Degree - Lexicographic ordering with an example

Let $X=\{x,y\}$ and $ x>y $ lexicographically. Let consider $X^*$ as a free monoid generated by $X$. We define a monomial ordering on $X^*$ which is compatible with the multiplication of words, due ...
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0answers
75 views

Is there any way to use Wronskian matrix to find out if a polynomial does not belong to an ideal?

Today I learned something about the Wronskian matrix. Can I use it to tell if some polynomial in $k[x_1,...,x_n]$ does not belong to a monomial ideal? I know if a polynomial belongs to an ideal it is ...
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1answer
590 views

Is reverse lexicographic order the same as graded reverse lexicographic order?

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse ...
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1answer
120 views

What does Hilbert series of monomial ideals describe?

I am trying to understand the point of Hilbert series of monomial ideals. I am confused because Macaulay has commands for hilbertSeries, hilbertPolynomial and hilbertFunction. What does Hilbert ...
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1answer
36 views

Demonstrations on the Simplicial complex of Graph

where I cannot understand $F\in\Gamma\land G\subseteq F\Rightarrow G\in\Gamma$. I would like to see an example about the simplicial complex of a graph such as a cycle graph $C_3$. What are ...
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46 views

A concrete example of an ideal $I\subseteq K[x_1,\ldots,x_n]$ and coprime polynomials $f,g$ such that $(I,f)\cap (I,g)\neq (I,fg)$

I know that in a polyomial ring $K[x_1,\ldots,x_n]$ over a field $K$, given a monomial ideal $I$ and two coprime monomials $f,g\notin I$, it holds $$(I,f)\cap (I,g)=(I,fg)$$ However, I've been ...
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1answer
89 views

Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
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1answer
94 views

Leading term ideal irreducible iff the ideal is irreducible?

Let $\mathbb{K}$ be a field. Given an ideal $I \subset \mathbb{K}[x_1,\dots, x_n]$ and a monomial order we can consider the ideal $LT(I) = (lt(f) \ | \ f\in I )$, where $lt(f)$ denotes the leading ...
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1answer
263 views

Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question is ...
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1answer
176 views

Show that some monomial ideal is primary

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary. I noticed that $$\sqrt{({X_{i_1}}^{a_1},...,{X_{i_k}}^{a_k})}=\sqrt{({X_{i_1}}^{a_1})+\cdots+({X_{i_k}}^{a_k})}=\...
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0answers
23 views

Show that $(X_{k_1}^{a_1},…,X_{k_s}^{a_s})$ is $(X_{k_1},…,X_{k_s})$-primary [duplicate]

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary, where $I$ is the ideal generated by the monomials $X_{k_1}^{a_1},...,X_{k_s}^{a_s}$ . I noticed that $$\sqrt{I}=\...
2
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1answer
495 views

Compute intersection of ideals in a polynomial ring

Consider the ideals $\mathfrak{p}_1=(x,y)$, $\mathfrak{p}_2=(x,z)$ and $\mathfrak{m}=(x,y,z)$ in $k[x,y,z]$. How to show that $\mathfrak{p}_1\mathfrak{p}_2=\mathfrak{p}_1\cap\mathfrak{p}_2\cap\...
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2answers
51 views

If $I,J$ are ideals in a polynomial ring over a field, how do I prove that $I = J$ if $\operatorname{in}_<(I)=\operatorname{in}_<(J)$?

If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering? ...
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1answer
322 views

Squarefree monomial ideals have a decomposition as the intersection of monomial prime ideals.

We've proven the following theorem in class: Every monomial ideal has a presentation $$I = \bigcap_{i=1}^m Q_i,$$ where each $Q_i = (x_{i_1}^{a_1}, \dots , x_{i_k}^{a_k})$. I've tried proving the ...
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1answer
161 views

How do you prove the ideal $I= (X^2, XY)$ has infinitely many distinct irredundant primary decompositions?

I have come up with the following two different decompositions of the ideal $I= (X^2, XY)$: $I = (X) \cap (X^2, Y)$ and $I = (X) \cap (X^2, XY, Y^2) = (X) \cap (X, Y)^2$. Can we generalize this ...
1
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1answer
94 views

I'm stuck with trying to construct a $K$-basis for the quotient of the polynomial ring $S/I$.

We were told in class that a $K$-basis for $S/I$ where $S=K[X_1, \dots , X_n]$ and $I$ a monomial ideal in $S$ is $W = \{X^a \in \mathrm{Mon}(S) \mid X^a \notin I\}$. I'm having difficulties ...
2
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2answers
148 views

How to check Cohen-Macaulayness?

Let $R=k[x,y,z]$. Consider the ideal $I=(x^2z^2,xyz,y^2z^4,y^4z^3,x^3y^5,x^4y^3)$. Is $R/I$ Cohen-Macaulay ? By definition it seems tough to solve this problem. Is there any other way to check this?
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1answer
300 views

Monomials and initial ideals

I am working on two questions for my Commutative Algebra assignment and am struggling to finish them. $1.$ Let $S=K[x_1,...,x_n]$, $I\subset S$ an ideal and $<$ a term order. I first showed that $...