A monomial ideal with given depth and dimension

Question

Assume we have two integers $a\leq b$. Is there a monomial ideal $I$ in a power series ring $R$ over a field such that $\dim(R/I)=b$ and $\mathrm{depth}(R/I)=a$?

Except for depth being at most the dimension, i do not know of any other restrictions in general for monomial ideals, hence the question. Dimension can be controlled easily since minimal primes of monomial ideals are well understood. On the other hand, i cannot think of a way to cook up an example with a specific depth. Any help would be appreciated. Thank you

It's a natural question whose answer isn't obvious. I suspect the techniques in this 2019 Question can be generalized to give certain constructions. See also this study of random monomial ideals which might produce a probabilistic proof. — hardmath, May 13 '24 at 22:24
Depth is a local condition . Do you mean depth at some localization at a maximal ideal is $a$? May be at the 'irrelevant' maximal ideal? — Mohan, May 14 '24 at 16:12

score 1 · Accepted Answer · answered May 14 '24 at 20:52

1

Let $R$ be the power series ring over a field in $x_0,x_1,\ldots,x_b$ and let $I=(x_0^2,x_0x_1,\ldots, x_0x_k)$. One checks that $\dim R/I=b$ and depth of $R/I=b-k$.

answered May 14 '24 at 20:52

Mohan

18,858

A monomial ideal with given depth and dimension

1 Answers1