Upper bound to the number of generators

Question

When defining noetherian ring/module there's no condition on the number of generators of ideals/submodules (apart from being finite). However, in some cases we can do better:

-A noetherian module over a field is a finite vector space, so every submodule can be generated with at most n elements.

-A maximal ideal of $\mathbb{K}[X_1,...,X_n]$ where $\mathbb{K}$ is algebraically closed can be generated, via Nullstellensatz, by exactly n generators.

What other examples are there where we can find a system of generators of bounded cardinality? What happens if we replace maximal ideal by prime ideal in the second example?

Answer 1

Understanding the number of generators is a very subtle problem. I will focus on your second question on ideals, since the first one is a bit broad. By a theorem of Foster-Swan, the problem is local.

There is no absolute upper bound even for a prime ideal of height $2$ in $k[[x,y,z]]$. In this paper, Moh gives a sequence of primes $(P_n)$ such that $\mu(P_n) =n+1$.

OK, so what to do next? One can ask if there are good bounds on $\mu(I)$ if $R/I$ is "nice". If $R/I$ is a complete intersection, then $\mu(I)$ is the height of $I$. In commutative algebra, the next level of "niceness" is being Gorenstein. In this paper Schoutens shows that if $I$ has height $2$ and $R/I$ Gorenstein, then there is a bound only depending on $R$.

If one goes down another notch, and only assume $R/I$ is Cohen-Macaulay, then Moh's examples show there are no hope for bounds independent of $I$. However, there are bounds that depends on invariants of $R/I$, such as the type or multiplicity.

Answer 2

When $(R, \mathfrak{m})$ is a one-dimensional Cohen-Macaulay local ring, then the multiplicity $\mathrm{e}(R)$ is the sharp upper bound for the number of generators $\mu_R(I)$ of ideals $I$ of $R$. It's sharp because it's also the stable value of $\mu_R(\mathfrak{m}^n)$ for $n$ large (this part doesn't need CM).