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.

understandit. $\endgroup$