Let \(X\) be a Noetherian scheme, let \(\mathcal F\) be a coherent \(\OO_X\)-module, let \(p\in X\), and let \(\kappa(p)=\OO_{X,p}/\mfm_p\). The fiber \(\mathcal F(p)\coloneqq\mathcal F_p/\mfm_p\mathcal F_p\) of \(\mathcal F\) at \(p\) is a finite-dimensional \(\kappa(p)\)-vector space.
Every lift to the stalk \(\mathcal F_p\) of a basis of \(\mathcal F(p)\) is a minimal generating set of \(\mathcal F_p\).
Given such a lift, there are an open neighborhood \(U\) of \(p\) and sections \(s_1,\ldots,s_n\in\mathcal F(U)\) with these germs at \(p\) that generate \(\mathcal F_q\) for every \(q\in U\).
If \(\mathcal F\) is locally free near \(p\), the sections \(s_1,\ldots,s_n\) of (b) can be chosen so that \(\OO_U^n\to\mathcal F|_U\), \((f_i)\mapsto\sum_if_is_i\), is an isomorphism; that is, a basis of the fiber of a vector bundle at \(p\) extends to a local frame on a neighborhood of \(p\).
The stalk \(\mathcal F_p\) is a finitely generated module over the local ring \(\OO_{X,p}\), so (a) is Nakayama's lemma for a local ring.
For (b), choose an open neighborhood \(V\) of \(p\) on which all the germs are represented by sections \(s_i\in\mathcal F(V)\), and let \(\mathcal C\) be the cokernel of \(\OO_V^n\to\mathcal F|_V\). Then \(\mathcal C\) is coherent with \(\mathcal C_p=0\), and the support of a coherent sheaf is closed, so \(\mathcal C\) vanishes on an open neighborhood \(U\subseteq V\) of \(p\).
For (c), shrink \(U\) to an affine open \(\Spec A\) containing \(p\) on which \(\mathcal F\) is free; its rank is \(\dim_{\kappa(p)}\mathcal F(p)=n\). Since \(\OO_U^n\to\mathcal F|_U\) is surjective and \(U\) is affine, the map \(A^n\to\mathcal F(\Spec A)\cong A^n\) is a surjective endomorphism of a finitely generated \(A\)-module, hence an isomorphism.