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.

Filter by
Sorted by
Tagged with
2 votes
1 answer
64 views

Isomorphism between polynomial rings with several variables

Here is a problem form my abstract algebra course : Let $R = \frac{\mathbb{Q}[u,v,w]}{(u^2v^2 - w^3)}$. Find finitely many monomials $x^{a_j}y^{b_j}$ in $S = \mathbb{Q}[x,y]$ such that $R \simeq \...
EtiBeranger's user avatar
1 vote
0 answers
49 views

Graded Betti numbers of monomial ideals and inclusions

Let $I$ and $J$ be two monomial ideals in some polynomial ring $S=k[X_1,...,X_n]$. Furthermore, assume that $G(I)\subseteq G(J)$, where $G(I)$ denotes the minimal set of monomial generators of $I$, ...
Diego Parodi's user avatar
3 votes
1 answer
43 views

Polynomial modulo a monomial [closed]

in an article I was reading it was mentioned that for some polynomial $p(x)$ in $\mathcal{Z}[x]$, the operation $p(x) \mod (x-r) = p(r)$, that is to compute the remainder of a polynomial modulo some ...
Jimakos's user avatar
  • 173
-7 votes
1 answer
43 views

Prove that every ideal of $\Bbb{Z}$ has the form $m\Bbb{Z}=\{mk, k\in \Bbb{Z}\}$ [closed]

Prove that every ideal of $\Bbb{Z}$ has the form $m\Bbb{Z}=\{mk, k\in \Bbb{Z}\}$ help me please, I really need that to do the homework
Thảo Phương's user avatar
2 votes
0 answers
76 views

Irreducibility of monomial ideals

So far, I have encountered two notions of irreducibility for ideals: We say that an ideal $I$ in a commutative ring $R$ is irreducible if it cannot be expressed as an intersection of two strictly ...
Diego Parodi's user avatar
0 votes
0 answers
42 views

Symmetric algebra represented as a quotient of a polynomial ring

I’m trying to understand the proof of Corollary 1.9 in “Binomial ideals” by David Eisenbud and Bernd Sturmfels. Notation: Let $S= k[x_1, \ldots, x_n]$ where $k$ is a field and the $x_i$ are ...
Artus's user avatar
  • 973
2 votes
1 answer
110 views

Is a monomial basis a Groebner basis?

Given a monomial ideal $$I = \langle x^{\alpha} \mid \alpha \in A \subset \Bbb Z^n_{\geq 0}\rangle,$$ I want to know if the rest of the polynomial division of $f \in \mathbb{K}[x_1,...,x_n]$ by the ...
user996159's user avatar
1 vote
0 answers
272 views

Example of initial ideal

I have a problem with understand some example, which I present below. Let ideal $I = (x_1^2 + 3x_1x_2, 2x_1^2 + x_2^2)$. The initial monomial of both generators is $x_1^2$. However, twice the first ...
Mathewg's user avatar
  • 21
2 votes
1 answer
168 views

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= \...
mathslover's user avatar
  • 1,482
2 votes
1 answer
191 views

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 ...
blancket's user avatar
  • 2,032
1 vote
1 answer
90 views

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 ...
user124697's user avatar
  • 1,737
1 vote
0 answers
88 views

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 ...
abacada's user avatar
  • 117
3 votes
1 answer
173 views

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 ...
krimas's user avatar
  • 31
0 votes
1 answer
391 views

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, ...
Brady Tim's user avatar
0 votes
0 answers
73 views

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 ...
simu tiyam's user avatar
1 vote
0 answers
43 views

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 $\...
hosseinsh's user avatar
3 votes
1 answer
130 views

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 "...
J.Ask's user avatar
  • 133
2 votes
1 answer
268 views

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 ...
lalaland's user avatar
0 votes
0 answers
43 views

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, $x^2$ ...
Amren Carver's user avatar
1 vote
0 answers
143 views

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 ...
ThirstForMaths's user avatar
1 vote
1 answer
220 views

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)...
one potato two potato's user avatar
1 vote
1 answer
170 views

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 = \...
user avatar
2 votes
1 answer
132 views

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 ...
user avatar
1 vote
1 answer
29 views

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 ...
Kilkik's user avatar
  • 1,867
1 vote
1 answer
72 views

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}...
Fernando Mauricio Rivera Vega's user avatar
2 votes
1 answer
70 views

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 ...
Dylan C. Beck's user avatar
3 votes
1 answer
158 views

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 ...
cqfd's user avatar
  • 12.1k
-2 votes
2 answers
178 views

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.
Math's user avatar
  • 49
0 votes
1 answer
91 views

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 ...
Pont's user avatar
  • 5,706
2 votes
1 answer
205 views

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 @>{\...
Antonio Ficarra's user avatar
0 votes
1 answer
34 views

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-...
User43029's user avatar
  • 1,235
1 vote
0 answers
248 views

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 ...
morrowmh's user avatar
  • 3,026
3 votes
0 answers
85 views

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}...
morrowmh's user avatar
  • 3,026
4 votes
2 answers
251 views

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 $m_1, \ldots, m_k$ ...
user5826's user avatar
  • 12k
1 vote
0 answers
107 views

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 ...
Mary's user avatar
  • 41
1 vote
0 answers
78 views

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\...
user521337's user avatar
  • 3,705
1 vote
0 answers
33 views

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 ...
takrp's user avatar
  • 147
0 votes
1 answer
117 views

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]...
ploosu2's user avatar
  • 8,445
1 vote
0 answers
22 views

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 ...
user's user avatar
  • 4,384
4 votes
0 answers
78 views

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^{...
user's user avatar
  • 4,384
2 votes
1 answer
154 views

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=\...
beginarray's user avatar
1 vote
1 answer
36 views

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 ...
sleeve chen's user avatar
  • 8,271
3 votes
1 answer
231 views

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,...
Randy Marsh's user avatar
  • 2,827
2 votes
1 answer
124 views

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 ...
mw19930312's user avatar
1 vote
2 answers
644 views

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 $(...
Harun rashid's user avatar
0 votes
1 answer
106 views

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^{\...
user2154420's user avatar
  • 1,441
0 votes
1 answer
105 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}...
sleeve chen's user avatar
  • 8,271
0 votes
1 answer
51 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&...
user102248's user avatar
  • 1,443
0 votes
1 answer
188 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 ...
uno's user avatar
  • 1,560
0 votes
1 answer
81 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 $(x,y,z) \in \mathrm{Ass}_R (R/J^2)$ ?
uno's user avatar
  • 1,560