Checking normality for quasi compact schemes

Question

Let $X$ be a quasi compact scheme. We know that any point on $X$ is a generization of a close point. Could someone possibly explain me why it then follows that to check if $X$ is normal, it suffices to check the normality at closed points? Thanks!

Answer 1

Because if you are an integrally closed ring $A$, so are all your localizations at prime (resp. maximal) ideals (and reciprocally). So for quasi-compact schemes, you can check normality at closed points.

Answer 2

0) This is a rather subtle question.
It is indeed true, as Robert pertinently remarks, that any localization of an integrally closed domain is integrally closed.
But what about schemes?

1) First of all, what is a normal scheme?
Answer: it is a scheme $X$ such that for each $x\in X$ the local ring $\mathcal O_{X,x}$ is a normal domain.
So it seems that it is enough to prove that such a point $x$ has a closed point $x_0\in \overline {\{x\}}$ in its closure , and then to conclude that $\mathcal O_{X,x}$, being a localization of $\mathcal O_{X,x_0}$, is also integrally closed.

2) However this idea alone doesn't lead to a correct proof because there exist (very counterintuitively!) schemes which have no closed point at all!

3) This is why we have the hypothesis that the scheme $X$ is quasi-compact: it is then a reasonably easy consequence that $X$ does have a closed point.
Since any closed subset of a quasi-compact space is also quasi-compact we can now prove the result : indeed given $x\in X$ the closure $\overline {\{x\}}$ is quasi-compact and thus contains a closed point $x_0$.
We can then legitimately apply the reasoning suggested in 1) .