# 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.

146 questions
Filter by
Sorted by
Tagged with
8 views

### 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 ...
18 views

70 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}...
40 views

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 $\Delta:=\left<x_1x_2x_3,x_3x_4x_5\right&... 0 votes 1 answer 142 views ### Primary decomposition of powers of a monomial (edge) ideal in a three variable polynomial ring Consider the ideal$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$. My ... 0 votes 1 answer 61 views ### Associated primes of the square of a monomial ideal Consider the ideal$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)$? 1 vote 1 answer 225 views ### On the height of a square-free monomial ideal in a five variable polynomial ring Consider the monomial ideal$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$5=\dim k[v,w,x,y,z]=\mathrm{ht}(I)+ \dim (k[v,w,x,y,z]/I)$,... 2 votes 0 answers 257 views ### 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$) such that$(... 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 ... 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 $<$ ...