# Questions tagged [monomial-ideals]

A monomial ideal in a polynomial ring is an ideal generated by monomials.

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### Projective dimension and certain subideals

My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...

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### Symmetric algebra of monomial ideal modulo a binomial ideal isomorphic to a quotient of a polynomial ring

Note: I posted this question on MSE but haven't received any response.
I’m trying to understand the proof of Corollary 1.9 in “Binomial ideals” by David Eisenbud and Bernd Sturmfels.
Notation: Let $S= ...

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### Is Koszul homology of a monomial ideal always generated by the "obvious" things?

Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form
$$r e_{...

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### Are there characteristic-dependent Betti numbers in characteristic not equal to two?

Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...

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### Reference request: Matroid cryptomorphisms for arbitrary monomial ideals

For a matroid $M$ let $C$ be the circuit ideal of $M$, that is, the Stanley-Reisner ideal of independence complex of $M$. Then there are simple ideal-theoretic operations that take $C$ to the facet ...

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### Infinitely many initial ideals for non-Artinian monomial orders?

Consider the polynomial ring $R=\mathbb Z[x_1,\ldots,x_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, ...

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### Reflection-invariant monomial ideals and Alexander duality

First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...

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### Betti numbers of a Cohen-Macaulay Module in small projective dimension

I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...

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### Gröbner bases of resultants and their monomial ideals

$\newcommand{QQ}{\mathbb{Q}}$
Consider the ring $R = \QQ[x, a_1,\ldots,a_m]$ for a certain integer $m$ and the homogeneous polynomial
$$
f = x^{m+1} + \sum_{i=1}^m a_i^i x^{m+1 - i}
$$
Now let
$$
...

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### Which ideals have standard Hilbert series?

Let $m$ and $d$ be two positive integers.
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...

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### The $k$ th symbolic power of a square free monomial ideal $\rm I$ is $\rm I^{(k)}= \cap_{p\in Min(I)}p^{k}.$

Let $\rm I$ is a square free monomial ideal in $K[X_1,\ldots,X_n].$ The $k$ th symbolic power of $\rm I$, denoted by $\rm I^{(k)}$ defined to be the intersection of all primary components of $\rm I^{k}...

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### Monomial algebras and depth

Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence.
Assume $...

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### Cut ideal of two graphs?

Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...

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### if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$

Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual.
Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$?
Is there a ...

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### ideal generated from a truncated "real radical-like" set are still real radical?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1...x_n$. $\mathbb{R}[X]_t$ is the truncated ring such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], deg(f) \leq ...

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### Vanishing ideal of a finite set of points does not have expected amount of cones in Gröbner fan

I am reading the paper A Gröbner fan method for biochemical network modeling.
In Chapter 4.3 (i.stack.imgur.com/h2O8B.png) they calculate the vanishing ideal of some tuples (input points of Series 1-...

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### Castelnuovo- Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be a graded ring and $S,T,U$ be monomials ideals.
$reg(S)=max\{j-i \backslash \beta_{i,j}(S) \neq 0\}$.
Assume $S+T=U$
prove \disprove : $reg(S+T^2) \leq reg(U^2)$.
We can ...

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### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is $x,x^2,...

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### Minimal free resolution of sum of ideals

Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the $\mathbb{...

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### Families of ideals with a given initial ideal

Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, ...

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### betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b):
can you help please?
...

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### Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},...,f_{m}...

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### Elementary proof that $\operatorname{Ass}(I^n)$ stabilizes for a monomial ideal $I$

For my bachelor thesis I'd like to have a short "elementary" proof that $\operatorname{Ass}(I^n)$ stabilizes for large $n$ if $I$ is a monomial ideal in a polynomial ring $K[x_1, \dots, x_r]$ over ...

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### Depth of ideals in a commutative ring

Let $I, J \subset S = k[x_1,\dots,x_n]$ be two monomial ideals and $k$ a field. If every element of $S$ which is $S/J$-regular is also $S/I$-regular is it true that depth$_S S/I \geq$ depth$_S S/J$ ?

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### How to show an ideal is zero-dimensional [closed]

I have the following past exam paper question, a similar sort of question seems to come up every year.. And I'm completely lost with it...
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by ...

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### Initial ideal of k-th power of an ideal

Hi,
Let $I$ be an ideal in a polynomial ring $S = k[x_1, \ldots, x_n]$, where $k$ is an algebraically closed field of characteristic zero. Fix a term order on
$S$ (e.g. a lexicographic order) and ...

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### How to find the generic initial ideal?

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra,p26-27
Let $f,g\in k[x_1,x_2,x_3,x_4]$ be a generic forms of degree $d$ and $e$, the generic initial ideal of $I=\langle ...

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### What does the d-slice of a weighted polynomial algebra look like?

This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...