The Stacks project

1. In an affine scheme if a finite number of points are contained in an open subset then they are contained in a smaller principal open subset. 2. Affine opens are cofinal among the neighborhoods of a given finite set of an affine scheme

Lemma 10.15.2 (Prime avoidance). Let $R$ be a ring. Let $I_ i \subset R$, $i = 1, \ldots , r$, and $J \subset R$ be ideals. Assume

  1. $J \not\subset I_ i$ for $i = 1, \ldots , r$, and

  2. all but two of $I_ i$ are prime ideals.

Then there exists an $x \in J$, $x\not\in I_ i$ for all $i$.

Proof. The result is true for $r = 1$. If $r = 2$, then let $x, y \in J$ with $x \not\in I_1$ and $y \not\in I_2$. We are done unless $x \in I_2$ and $y \in I_1$. Then the element $x + y$ cannot be in $I_1$ (since that would mean $x + y - y \in I_1$) and it also cannot be in $I_2$.

For $r \geq 3$, assume the result holds for $r - 1$. After renumbering we may assume that $I_ r$ is prime. We may also assume there are no inclusions among the $I_ i$. Pick $x \in J$, $x \not\in I_ i$ for all $i = 1, \ldots , r - 1$. If $x \not\in I_ r$ we are done. So assume $x \in I_ r$. If $J I_1 \ldots I_{r - 1} \subset I_ r$ then $J \subset I_ r$ (by Lemma 10.15.1) a contradiction. Pick $y \in J I_1 \ldots I_{r - 1}$, $y \not\in I_ r$. Then $x + y$ works. $\square$

Comments (9)

Comment #1506 by jojo on

Suggested slogan "In an affine scheme if a finite number of points is contained in an open subset then they are contained in a smaller principal open subset".

Comment #1544 by jojo on

Suggested slogan: In an affine scheme if a finite number of points is contained in an open subset then they are contained in a smaller principal open subset

Comment #1545 by jojo on

Oops sorry I tried the slogan generator and it did the same thing as my comment.

Comment #4333 by comment_bot on

The slogan of this tag is displayed as "None". Is this intentional?

Comment #4334 by on

@#4333: No, this is not intentional. This is an issue which might have been caused by the server move (although it's unclear to me how), and I made a GitHub issue at https://github.com/gerby-project/gerby-website/issues/141.

I will look into it today or tomorrow. For now you can go to the section page, where for some reason the slogan is still displayed correctly.

Comment #5495 by comment_bot on

The slogan does not match the content and, anyway, is too long. Also, the condition in the slogan holds tautologically: any finite set of points of any space is contained in an open, namely, the space itself.

Comment #5499 by on

Are you sure you read the slogan correctly? Note the word "smaller" in the conclusion. But I do agree this slogan isn't optimal. I would say the slogan for this result is simply "Prime avoidance". Anybody else want to comment?

Comment #5506 by comment_bot on

OK, I now see that the slogan could be understood the right way, but to me this interpretation of "if" is not the default one: it sounds like a condition that the given finite number of points is supposed to satisfy.

Anyway, here is another possible slogan: "Affine opens are cofinal among the neighborhoods of a given finite set of an affine scheme"

There are also:

  • 5 comment(s) on Section 10.15: Miscellany

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View Lemma 10.15.2 as pdf