There is a lemma in commutative algebra:
- Let $\mathfrak{a}_1, \dotsc, \mathfrak{a}_n$ be ideals such that $\mathfrak{a}_n \cap \dotsb \cap \mathfrak{a}_n$ is contained in a prime ideal $\mathfrak{p}$. Then $\mathfrak{a}_i \subseteq \mathfrak{p}$ for some $i$.
- Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_n$ be prime ideals such that $\mathfrak{p}_1 \cup \dotsb \cup \mathfrak{p}_n$ contains an ideal $\mathfrak{a}$. Then $\mathfrak{p}_i \supseteq \mathfrak{a}$ for some $i$.
It has a nice proof which is a little tricky (given in Atiyah/MacDonald chapter 1).
This lemma has fallen out of my mind a couple of times (along with its proof), and I'm trying to find a way to remember it better. Often I remember things better if I have a geometric picture of what's going on, or if I can generalize the statement somehow.
I asked a friend about a geometric version of it. His thoughts were that the first one relates to the fact that any subvariety of an irreducible variety must have positive codimension, and finite unions of these can't comprise the whole space. The second, he said, didn't have a good geometric picture, because the union of ideals is not a geometric operation.
Does anyone have thoughts on
- Generalizations of this lemma
- Other proofs using the same style of argument
- Intuitions for it
- Where it is used (probably a lot of places, so perhaps extra-salient ones)