A revised version of the notes is now posted in the usual place.

I am continually learning more about how rich and complex the notion of associated points is. I had first understood them in terms of primary decomposition. My initial presentation was quite algebraic, and geometrically unmotivated. I later realized that associated points could be better understood without primary ideals, and the result would be streamlined and shorter. The result was in the version originally posted on this site (and that remained in the notes until the new version). I was pleased with myself: people could get through associated points without too much sweat, although without much motivation. (Feedback from readers and students confirmed this: people found this section more obscure than I would have liked.)

Just over a year ago, Matthew Emerton vigorously argued in this comment that there was a much better geometric point of view. My experience in the past is that when Matt makes a point like this, it causes a revolution in how I think about the topic, and indeed it happened again. (Charles Staats joined the discussion as well.)

As I’m about to discuss associated points in this year’s course, I’ve had a chance to carefully think over Matt’s point of view. I’m not surprised that he convinced me; but I am surprised at how big a difference his perspective made to me. (I also find striking that this is my third point of view on associated points, and I can even see a case for a fourth or a fifth depending on how you think. As one example, primary ideals are certainly absolutely central from many points of view.)

Section 6.5 now attempts to get across this perspective, and I offer it somewhat tentatively, because I may be missing some insights that will make it cleaner still.

Here is the current exposition. As always, the most important things for a reader/learner/expert to know are properties of a concept, not the definitions or proofs. I state the key properties first; they are quite different than the key properties I had in the previous incarnation. As I now see it, the key properties are this. Let $A$ be a Noetherian ring (for now), and $M$ a finitely generated $A$-module.

Following (my interpretation of) Matt’s inspiration: the most important property is:

(A) The associated points of $M$ are precisely the generic points of irreducible components of the supports of sections of $\tilde{M}$ (elements of $M$).

This leads to lots of useful properties by purely geometric thinking (as Matt points out). We could even take it as the definition, but in the rigorous development, I don’t. (Side point: a variant of this is the statement that the associated points of $M$ are precisely the generic points of supports that happen to be irreducible. I found this less useful, but perhaps I’m missing some insights. It’s possible Matt had this definition in mind.)

There are two other “first principles” to keep in mind.

(B) There are a finite number of associated points.

(C) A function is a zero-divisor if and only if it vanishes at an associated point.

I couldn’t get (B) and (C) to follow cheaply from (A). But lots of great properties follow from these (especially (A), sometimes with the help of (B)), including:

1. Generic points of irreducible components are associated.
2. The map from $M$ to the product of stalks/localizations at the associated points is injective.
3. The support of $m \in M$ is the closure of those associated points where it is supported.
4. The nonreduced locus of Spec $A$ is the closure of the nonreduced associated points.
5. The notion of associated points behaves well with respect to localization, so for example we can define associated points of (coherent sheaves on) locally Noetherian schemes.

In order to establish (A)(C), I need some algebra, which I do in a series of exercises. I’m not surprised that algebra is necessary, because Noetherianness (of the “sheaf”, not just the topology) needs to be used. But perhaps the exposition can be “geometrized” further, to make it more geometrically natural. (Any suggestions would be appreciated — not just for the notes, but for me!)

In particular, the definition of associated prime I take is a prime that is the annihilator of some element of $M$. I show (B) and (C) directly, as well as #2 and #5, which I then use to show (A).

Any comments on the exposition would be greatly appreciated. Am I missing some fundamental insights that would simplify things? Is the exposition readable to someone seeing it for the first time? Are the problems gettable by people seeing these ideas for the first time? I’ll get some feedback from my class, but they are not representative of a possible readership (as many of them know a ridiculous amount).

(I’m curious how much of this carries over out of the Noetherian or integral setting. If the various definitions then differ, which is the “right” one? But this is a secondary issue; I haven’t thought about it much, and it also is less relevant for most learners.)