Short version: Why is the normalization of a separated scheme again separated?
Long version: I am trying to understand the construction of the normalization of an irreducible variety (or scheme if you want). In the affine case one just takes the spectrum of the integral closure of the coordinate ring, this is okay for me. For arbitrary varieties one takes an affine cover normalizes the pieces and glues them together, this is also okay to me. However I don't see why the result is separated. We have an affine cover with affine intersections for free, but I don't see why the coordinate rings of the intersections are generated by coordinate rings of the pieces I intersect. Maybe I am missing a simple fact in commutative algebra.
Any explanation or a reference where separatedness ist proven?