# Questions tagged [monomial-ideals]

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

28 questions
Filter by
Sorted by
Tagged with
1 vote
75 views

• 141
272 views

• 630
308 views

### 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 ...
• 457
1 vote
256 views

1 vote
91 views

### 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 ...
• 143
1 vote
82 views

### 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 ...
• 1,389
154 views

• 1,960
132 views

283 views

### 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 ...
• 21
525 views

### 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$_S S/J$ ?
• 187
2k views

### 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 ...
1 vote
490 views

### 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 ...
• 883
2k views

### 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 ...
• 371