Recall Chow's lemma:
Chow's Lemma: If $X$ is a scheme that is proper over a noetherian base $S$ then there exists a projective $S$-scheme $X'$ and a surjective $S$-morphism $f : X'\to X$ that induces an isomorphism $f^{{-1}}(U) \simeq U$ for some dense open $ U\subset X$.
I just looked at the proof on wikipedia (which presumably the one from EGA II) and found a slightly bothering detail. It starts by reducing to the irreducible case with the following argument:
Argument: Let $U_i \subset X_i$ be the open dense subsets in each irreducible component which satisfy the lemma. Then $U := \coprod_i (U_i -\bigcup_{j \ne i} X_j)$ is open dense in $X$ and satisfies the lemma.
If we interpret dense as dense in the zariski topology then everything works fine throughout. But if we want it to mean scheme theoretically dense there's a problem with the argument as their might be an embedded point in $U_i \cap X_j$ (consider two lines intersecting at a double point).
Is there a way to "fix" the argument so that Chow's lemma will be true for dense = scheme theoretically dense?
If not is there a simple example of a non-reduced scheme for which this "strong" Chow's lemma fails?