Hartshorne Exercise II.7.1.
Let $(X,\OO_X)$ be a locally ringed space, and let $f:\mathcal{L}\ra\mathcal{M}$ be a surjective map of invertible sheaves on $X$. Show that $f$ is an isomorphism. [Hint: Reduce to a question of modules over a local ring by looking at the stalks.]
Since $f$ induces a commutative square on every stalk,
$$f:\mathcal{L}_p\ra\mathcal{M}_p$$
$$\downarrow\quad\quad\:\downarrow$$
$$\OO_{X,p}\ra\OO_{X,p}$$,
where the columns are isomorphisms and the bottom arrow is a surjection as $\OO_{X,p}$-modules, we reduce immediately to proving the following standard fact about modules over commutative rings. That is, let $M$ be a nonzero finitely generated module over a local ring $R$. Then any surjective $R$-module homomorphism $f:M\ra M$ is an isomorphism. To show this we will use Nakayama’s lemma. Consider $M$ as as an $R[X]$-module by $X\cdot m=f(m)$. We see that $(X)\cdot M=M$ so by Nakayama’s lemma there is some polynomial $P(X)$ with $(1-P(X)X)\cdot M=0$. But then $P(f)$ is an inverse to $f\in \op{End}_R(M)$, and therefore $f$ is an isomorphism. This actually proves the problem for locally free sheaves of the same rank.
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