Favourite applications of the Nakayama Lemma

Solution 1:

Let $A$ be a commutative ring and $M$ a finitely generated $A$-module. Then every surjective endomorphism $f:M\to M$ is injective .
This result (due to Vasconcelos) surprizingly holds in complete generality, without any noetherianness assumption, and crucially uses Nakayama in its proof:

The trick is to consider $M$ also as an $A[X]$-module via the multiplication $P(X)\cdot m=P(f)(m)$, so that for example $(X^3-X)\cdot m=f^3(m)-f(m)$.
The surjectivity asssumption translates into $M=IM$, where $I$ is the ideal $I=(X)\subset A[X]$.
Nakayama then says that for some $i=Q(X)X\in I$ we have $m=i\cdot m$ for all $m\in M$.
[Needless to say, since $M$ is finitely generated over $A$, it is a fortiori finitely generated over $A[X]$ so that Nakayama may legitimately be invoked.]
And now the injectivity of $f$ follows: if $f(m)=0$ we have successively $$m=i\cdot m=Q(X)X\cdot m=Q(f)(f(m))=Q(f)(0)=0$$ so that $m=0$, which proves the injectivity of $f$.

Solution 2:

I never remember where it is used, but I did manage to retain one simple application that follows directly from the statement.

If $I$ is a finitely generated ideal of a ring (with identity) such that $I^2=I$, then $I$ is a ring with identity.

Maybe I am just too surprised by theorems which conclude that a rng has an identity :)

Solution 3:

Here is a result in Algebraic Geometry that can be an application of Nakayama:

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent sheaf. Then $\mathscr{F}$ is invertible if and only if there exists some coherent sheaf $\mathscr{G}$ such that $\mathscr{F}\otimes\mathscr{G}=\mathcal{O}_X$.