The Stacks project

Lemma 10.63.20. Let $k$ be a field. Let $S$ be a finite type $k$ algebra. If $\dim (S) > 0$, then there exists an element $f \in S$ which is a nonzerodivisor and a nonunit.

Proof. By Lemma 10.63.5 the ring $S$ has finitely many associated prime ideals. By Lemma 10.61.3 the ring $S$ has infinitely many maximal ideals. Hence we can choose a maximal ideal $\mathfrak m \subset S$ which is not an associated prime of $S$. By prime avoidance (Lemma 10.15.2), we can choose a nonzero $f \in \mathfrak m$ which is not contained in any of the associated primes of $S$. By Lemma 10.63.9 the element $f$ is a nonzerodivisor and as $f \in \mathfrak m$ we see that $f$ is not a unit. $\square$

Comments (7)

Comment #7753 by Kentaro Inoue on

I think this lemma can be proved more easily. Since has infinite maximal ideals, there exists a maximal ideal which is not equal to any associated prime ideal. We can take an element which is not contained in any associated prime ideal. Then is a desired one.

Comment #8319 by Zichen Lu on

Why does the ring S have infinitely many maximal ideals? Lemma 0ALW only implies that S has infinitely many prime ideals, but not maximal ideals.

Comment #8320 by on

Thanks for pointing out this insufficient argument. Of course this is easily fixed by saying that an infinite Jacobson space has infinitely many closed points; I will do this when I next go through all the comments again. Does anybody have another suggestion for fixing this?

Comment #8324 by Rijul Saini on

I just wanted to remark that by Noether normalization, it is enough to see that the affine line over k has infinitely many closed points.

There are also:

  • 14 comment(s) on Section 10.63: Associated primes

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GEC. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 10.63.20 as pdf