Arithmetic rank and height

Question

I've been looking for an example of an ideal whose height and Arithmetic rank is different.

I think that it must differ just in 1, but i can't imagine an example.

$\mathbb{ara}(I)=\mathbb{ht}(I)+1$

Answer 1

The difference ${\rm ara}(I) - {\rm ht}(I)$ can be arbitrarily large even for monomial edge ideals, i.e. monomial ideals generated in degree two. In fact, for a squarefree monomial ideal $I$ generated in degree 2 in a polynomial ring $R$, the following inequalities hold: $$ {\rm ht}(I) \leq {\rm bight}(I) \leq {\rm pd}(R/I) \leq {\rm ara}(I), $$ where ${\rm bight}(I)$ is the big height of $I$, the maximum height of the minimal primes of $I$.

Let $n \geq 2$ and consider the ideal $I_n = (x_0 x_i : i=1,\dots,n)$ in $R=K[x_0,\dots,x_n]$. Now, it can be easily shown that $1 = {\rm ht}(I_n) < {\rm bight}(I_n) = {\rm pd}(R/I_n) = {\rm ara}(I_n) = n$.