Locally Free Sheaves If $X$ is a locally noetherian scheme and $F$ is a coherent sheaf, I want to show the following equivalence:
$F$ is locally free iff its stalk is a free $O_{X,p}$-module for every $p$ in $X$.
=> follows from the definition of locally free. 
<= is difficult for me: I don't see how to combine finite-type condition on $F$ with locally noetherian property. 

As the name indicates, "locally free" is a local concept.
So we can assume that $X=Spec(A)$, the  affine scheme associated to the noetherian ring $A$, and $F=\tilde M$, the coherent sheaf associated to the finitely generated $A$-module $M$. 
The sheaf $F=\tilde M$ is locally free if and only if the module $M$ is projective.
And a finitely generated   module $M$ over a noetherian ring $A$ is projective if and only if all its localizations $M_{\mathfrak p} \;  ( \mathfrak p \in Spec(A)$ are $A_{\mathfrak p}$-free modules.
Since at  a point $p\in  X$ corresponding to the prime $\mathfrak p \in Spec(A)$ we have $\mathcal O_{X,x}=A_{\mathfrak p}$ and $ F_p=M_{\mathfrak p}$,  we see that indeed freeness of all stalks of $F$ implies local  freeness of the given coherent sheaf $F$.
