I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as follows: If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain finite separable extension of the field $R(X)$ of rational functions on $X$, then $X'\to X$ is a finite morphism.
I have an issue with the proof of this statement in the book. Consider a particular case when both $X$ and $X'$ are affine and $A=O(X)$ is an integral domain, integrally closed in the fraction field $K$ of $A$.
We are given a finite separable extension $L$ of $K$, then by definition, $B=O(X')$ is an integral closure of $A$ in $L$, and we have to show that $B$ is finite over $A$.
The author makes a reference to Prop 5.17 of the Atiyah-MacDonald Commutative Algebra book, which only claims that there is a basis $v_1,v_2,\ldots,v_n$ of $L$ over $K$ such that $B$ is contained in the $A$-span of $v_1,v_2,\ldots,v_n$, that is $B$ is an $A$ sub-module of a free finitely generated $A$-module. However, in general, this does not guarantee that $B$ itself is a finitely generated $A$-module. Is it indeed a flaw in the argument or I do not see something?
