What is the universal property of normalization? I'm looking for an answer something like



> 
> If X is a scheme and Y→X is its
>  normalization, then the morphism
>  Y→X has property P and any other
>  morphism Z→X with property P
>  factors uniquely through Y.
> 
> 
> 



What is the universal property of normalization?
I've realized that my answer is wrong. Here's the right answer: if $Z$ is a normal scheme and $f: Z \to X$ is a morphism such that each associated point of $Z$ maps to an associated point of $X$, then $f$ factors through $n$.


A counterexample that shows why what I said previously doesn't work: let $f$ be the inclusion of the node into the nodal curve. There is no unique lift of $f$ to the normalization of the nodal curve.


What's going on here: taking the total ring of fractions is not a functor for arbitrary morphisms of reduced rings. You need a morphism such that no NZD gets mapped to a ZD, which is equivalent (for Noetherian rings) to saying that the preimage of any associated prime is an associated prime.


