TITLE: How to memorise (understand) Nakayama's lemma and its corollaries?
QUESTION [201 upvotes]: Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe around 10 times) in my life. 
But for some reason I just cannot get this lemma, i.e. I have tendency to forget it. Last time this happened just a couple of days ago, in the book of Shafarevich (Basic Algebraic geometry in 1.5.3.) This lemma is used to prove that for finite maps between quasiprojective varieties the image of a closed set is closed, and again this lemma sounded as something foreign to me (so again I went through the proof of the lemma)... 
Question. Is there a path to get some stable understanding of Nakayama's lemma and its corollaries? I would be especially happy if there were some geometric intuition underlying this lemma. Or some geometric example. Or maybe there is a nice article of this topic? Some mnemonic rule? (or one just needs to get used to the lemma?)

REPLY [12 votes]: I just came up with a geometric interpretation for Nakayama's lemma, and I'm surprised that no one here has already mentioned it!
The statement is the following: given a ring $A$, an ideal $I$, and a finitely generated $A$-module $M$, if $IM=M$ then we can find an element $a\in I$ such that $(a-1)M=0$.
In the geometric picture, we see $A$ as functions on a space $X$ (the spectrum $\mathrm{Spec}(A)$), with $I$ corresponding to functions that vanish on a closed subset $Z$ (the subset $V(I)$); $M$ is the module of global sections for a "vector bundle" $\mathcal F$ (the sheaf of modules $\widetilde M$). The elements of $IM$ represent those global sections vanishing along $Z$, so the condition $IM=M$ says that these are all the sections possible. In other words, $\mathcal F$ is identically zero along $Z$, and so its support $\mathrm{Supp}(\mathcal F)$ must be disjoint from $Z$. Now the conclusion of Nakayama's lemma simply affirms the existence of a "bump function" $a$, that is, a function being $0$ along $Z$, and $1$ along the support of $\mathcal F$: i.e. it lies in $I$ and it acts on $M$ like $1$!
If one works out the details of commutative algebra,

the condition "$\exists\ a\in I$ s.t. $(a-1)M=0$" is equivalent to $V(I)\cap V(\mathrm{Ann}(M))=\emptyset$ (the "partition of unity");
the support $\mathrm{Supp}(M)$ is contained in $V(\mathrm{Ann}(M))$;
for finitely generated module $M$, the condition $IM=M$ implies $V(I)\cap\mathrm{Supp}(M)=\emptyset$, moreover $\mathrm{Supp}(M)$ can be shown to coincide with the closed subset $V(\mathrm{Ann}(M))$.


When $M$ is not finitely generated, $\mathrm{Supp}(M)$ does not behave well, so we have courterexamples

$IM=M$ does not imply $V(I)\cap\mathrm{Supp}(M)=\emptyset$: the classical counterexample $A=\mathbf Z_{(p)}$ with its maximal ideal, and $M=\mathbf Q$;
$\mathrm{Supp}(M)$ can be non-closed: $A=\mathbf Z$, $I=2\mathbf Z$, and $M=\bigoplus_{p\ge3}\mathbf F_p$;
$\mathrm{Supp}(M)$ can be closed yet different from $V(\mathrm{Ann}(M))$: $A=\mathbf Z$, $I=2\mathbf Z$, and $M=\bigoplus_{n\ge1}\mathbf Z/3^n\mathbf Z$.


So, to conclude, Nakayama's lemma (in the above form) says that the support of a finitely generated $A$-module $M$ can be identified with the closed subset $V(\mathrm{Ann}(M))$, and given any closed subset $Z=V(I)$ disjoint from it, we can find a "bump function" separating the two.
In real life, however, we are usually just using another (weaker) property of the closedness: the support is closed under specialization. Since for an $A$-module $M$ to be non-zero, it must at least have some support, and by specialization, it must be supported at least at one closed point (corresponding to a maximal ideal). Therefore, if we can verify that $M$ is not supported at any of the closed points, (which is provided, say, by the condition $IM=M$ for $I$ contained in the Jacobson radical $J(A)$), then $M$ must be zero.