Hartshorne Exercise II.5.8.
Again let X be a noetherian scheme, and $\mathcal{F}$ a coherent sheaf on X. We will consider the function,
$$\varphi(x)=\op{dim}_{k(x)}\mathcal{F}_x\otimes_{\OO_x}k(x)$$,
where $k(x)=\OO_x/\mathfrak{m}_x$ is the residue field at the point $x$. Use Nakayama’s lemma to prove the following results.
(a) The function $\varphi$ is upper semi-continuous, i.e., for any $n\in\Z$, the set $\{x\in X|\varphi(x)\ge n\}$ is closed.
(b) If $\mathcal{F}$ is locally free, and $X$ is connected, then $\varphi$ is a constant function.
(c) Conversely, if $X$ is reduced, and $\varphi$ is constant, then $\mathcal{F}$ is locally free.
(a) We will show that the complement of this set is open. Indeed, let $x\in X$ with $\varphi (x)=m<n$. Let $x\in U=\spec A$ be an open neighborhood, $\mathcal{F}|_U\cong \tilde M$ for a finitely generated $A$-module $M$. By Nakayama’s lemma, $\mathcal{F}_x$ can be generated by m elements. Following our argument in 5.7 (a), there is an element $f\in A-\mathfrak{p}$ and a homomorphism $A[\frac 1f]^{m}\xrightarrow{\varphi} {M[\frac{1}{f}]}$ with $\op{coker}_(\varphi)\mathfrak{p}=0$. We can therefore find an element $g\in A-\mathfrak{p}$ that gives a surjective homomorphism of $\OO_{\{fg\ne 0\}}$-modules, $\OO_{\{fg\ne 0\}}^{\oplus m} \ra \mathcal{F}|_{\{fg\ne 0\}}$. It follows that for every point in $\{fg\ne0\}$ the vector space $\mathcal{F}_x\otimes{\OO_{X,x}}k(x)$ has dimension at most m.
(b) Suppose $\mathcal{F}$ is locally free. Let $x\in X$ and write $\varphi(x)=n$. I claim that the set $S:={x\in X|_\varphi(x)=n}$ is clopen. Then (b) is an immediate consequence. It is clear that for any $y\in X$, $\op{rank}(\mathcal{F}_y)=\varphi (y)$. Moreover, as we saw in 5.7, for any $y\in X$ there is a whole open neighborhood $y\in U$ over which $\op{rank}(\mathcal{F}|_U)=\op{rank}(\mathcal{F}_y)$. Thus, if $y\in S$ then $\varphi|_{U}\equiv n$ so $S$ is open, and if $y\notin S$, say $\varphi(y)=m\ne n$ then likewise $\varphi|_{U}\equiv m$, so $S$ is closed. This proves the claim.
(c) By Nakayama’s lemma, for every $\mathfrak{p}\in X$, $\mathcal{F}_\mathfrak{p}$ can be generated by say $n$ elements. Assume that $n\ge 1$, else $\mathcal{F}=0$ is trivially free. Also for every point $\mathfrak{p}\in X$ there is an open neighborhood $\mathfrak{p}\in U$ and a short exact sequence of $\OO_U$-modules,
$$ 0\ra\mathcal{K}\xrightarrow{\psi}\OO_U^{\oplus n}\xrightarrow{\phi}\mathcal{F}|_U\ra 0.$$
At every stalk we can apply $\_\_\otimes_{\OO_{X,\mathfrak{p}}}k(\mathfrak{p})$ to find that $\op{im}\psi_\mathfrak{p} =0$ (note that $k(\mathfrak{p})$ is not necessarily a flat $\OO_{X,\mathfrak{p}}$-module). In cash,
$$\psi_\mathfrak{p}:\op{ker}\phi _\mathfrak{p}\otimes k(\mathfrak{p})\ra k(\mathfrak{p})^n\:\text{ sends }\:\sum_{i=1}^na_i\otimes 1\mapsto \sum_{i=1}^n \bar a_i=0.$$
The sum on the right is $0$ iff $\bar a_i=0\:\:\forall i$ because $\op{ker}\phi _\mathfrak{p}\ra\OO_{X,\mathfrak{p}}^{\oplus n}$ is an injection. It follows that
$$\op{ker}\phi_{\mathfrak{p}}\subset (\mathfrak{p}\OO_{X,\mathfrak{p}})^n\quad \forall \:\mathfrak{p}\in U.$$
WLOG $U$ is affine, so momentarily using the same notation for maps on global sections (like we have done in (a)), this implies that
$$\op{ker}\phi\subset (\mathfrak{N}_{\OO_U(U)})^n=0,$$
because $X$ is reduced. We conclude that $\phi$ is an isomorphism, $\OO_U^{\oplus n}\cong\mathcal{F}|_U$, as desired.
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