Here is the statement of a version of Nakayama's lemma in Mumford's red book. Let $X$ be a noetherian scheme, $F$ a coherent $O_X$-module and $x \in X$. If $U$ is a neighbourhood of $x$ and $a_1, \ldots a_n \in \Gamma(U,F)$ have the property: the images $\overline{a_1}, ..., \overline{a_1}$ generate $F_x \otimes_{O_{X,x}} k(x)$ then there exists a neighbourhood $U_0 \subset U$ of $x$ such that $a_1, ..., a_n $ generate $F |_{U_0}$.
There are two things I don't understand in this statement, which I haven't been able to deduce looking at the proof.
-"$\overline{a_1}, ..., \overline{a_1}$ generate $F_x \otimes_{O_{X,x}} k(x)$" over $O_{X,x}$ or $k(x)$?
-what does it mean by $a_1, ..., a_n $ generate $F |_{U_0}$?
thank you.
The map $O_{X,x} \to k(x)$ is surjective, so generated over $O_{X,x}$ or $k(x)$ is the same thing.
For the second question, it means that the images of $a_1, ..., a_n $ in $F_y$ are generators for this $O_{X,y}$-module for each $y\in U_0$.