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?

  • $\begingroup$ Your examples seem to me to be of distinctly different flavors. In the first, the number of generators of a submodule of an n-generated module is at most n, but the ring (the field) remains fixed. In the second, the number of generators of maximal ideals is at most $n$, but the ring is varying with $n$. $\endgroup$ – Graham Leuschke Nov 19 '10 at 0:41
  • $\begingroup$ Have you looked at ideals such as (x,y)^n in k[x,y]? $\endgroup$ – roy smith Nov 19 '10 at 1:45

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.


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).

  • $\begingroup$ This is a very nice example when absolute bound exists! $\endgroup$ – Hailong Dao Nov 19 '10 at 4:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.