What is the universal property of normalization?

Question

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.

Answer 1

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.

Answer 2

If $X$ is a variety, the normalization $X'\to X$ is the maximal finite birational map to X, and the minimal dominant map from a normal variety to X, where maximal means any other finite birational map $Y\to X$ fits in a unique diagram $X'\to Y\to X$, and minimal means any other dominant map $Z\to X$ from a normal variety $Z$ fits in a unique diagram $Z\to X'\to X$.

In particular a variety is normal if it admits no non trivial finite birational maps. This gives you immediately that nodal curves are not normal, and neither are any varieties obtained by identifying points.

The meaning of normality is explained extremely clearly on page 181, in Mumford's unpublished second volume Algebraic geometry book, revised and edited now by Oda, and available online. http://www.math.upenn.edu/~chai/624_08/math624_08.html

The main geometric point is that normal points are locally irreducible, or "unibranch". The more delicate analytic aspect is the Hartog's extension theorem.

The key is Zariski's main theorem: I quote from Mumford-Oda:

"6. Zariski’s Main Theorem A second major reason why normality is important is that Zariski’s Main Theorem holds for general normal schemes. To understand this in its natural context, first consider the classical case: $k = C$, $X$ a $k$-variety, and $x$ is a closed point of $X$. Then we have the following two sets of properties:

N1) $X$ formally normal at $x$, i.e., $\mathcal{O}_{x,X}$ an integrally closed domain.

N2) $X$ analytically normal at $x$, i.e., $\mathcal{O}_{x,X,an}$, the ring of germs of holomorphic functions at $x$, is an integrally closed domain.

N3) $X$ normal at $x$.

N4) Zariski’s Main Theorem holds at $x$, i.e., $\forall \; f : Z \to X$, $f$ birational and of finite type with $f^{−1}(x)$ finite, then $\exists \; U \subset X$ Zariski-open with $x \in U$ and res $f : f^{−1}U \to U$ an isomorphism.

U1) $X$ formally unibranch at $x$, i.e., Spec $\mathcal{O}_{x,X}$ irreducible.

U2) $X$ analytically unibranch at $x$, i.e., Spec ($\mathcal{O}_{x,X,an}$) irreducible, or equivalently, the germ of analytic space defined by $X$ at $x$ is irreducible.

U3) $X$ unibranch at $x$, i.e., if $X'$ = normalization of $X$ in $R(X)$, $\pi : X' \to X$ the canonical morphism, then $\pi^{−1}(x)$ is a single point.

U4) $X$ topologically unibranch at $x$ — cf. Part I [76, (3.9)].

U5) The Connectedness Theorem holds at $x$, i.e., $\forall f : Z \to X$, $f$ proper, $Z$ integral, $f(\eta Z) = \eta X$ and $\exists U \subset X$ Zariski-open with $f^{−1}(y)$ connected for all $y \in U$, then $f^{−1}(x)$ is connected too.

6.1. I claim: i) all properties N are equivalent,

ii) all properties U are equivalent,

iii) $N \Rightarrow U$."

The reference [76] is to Mumford's Alg Geom I, Complex projective varieties. Compare also the discussion of ZMT in his red book.

Answer 3

Normalization is right adjoint to the inclusion functor from the category of normal schemes into the category of reduced schemes. In other words, if $n:Y\rightarrow X$ is the normalization of $X$ and $f:Z\rightarrow X$ is any morphism where $Z$ is a normal scheme, then $f$ factors uniquely through $n$.

Answer 4

I guess that you may not be worried about this anymore, but Prop. 12.44 in Görtz and Wedhorn's book answers your question (on integral schemes): For every integral and normal scheme Y' and every dominant morphism$ f: Y' \to X $ there exists a unique morphism $g: Y' \to Y$ such that $n\circ g = f$.

Answer 5

I think it's okay if you rephrase like this: let X be a scheme with ring of fractions R. Then the normalization of X is the universal example of a normal scheme with ring of fractions R mapping to X.

Answer 6

While this comes pretty late I wanted to add that normalization has a somewhat stronger universal property than what is mentioned in the other answers:

(N) Let $f: Z \to X$ be a morphism between integral, excellent schemes with $Z$ normal such that the image contains a normal point of $X$. Then $f$ lifts uniquely to the normalization $Y \to X$.

If the cited normal point is the generic point of $X$, one gets back the criterion demanding dominance (ND), as in Giulia's answer. However it also encompasses situations like a normal curve $Z$ on a singular surface $X$ not contained in the singular locus, which also lifts uniquely to the normalization.

To justify (N) one uses (ND) to reduce to $Z$ being the normalization of a closed integral subscheme $R \subset X$ whose generic point $\eta$ is normal in $X$. Denote by $R'$ the reduced closure of $\{\eta\}$ in the normalization of $R \times_X Y$. Because $\eta$ is a normal point of $X$, this is the unique component mapping dominantly to $R$. In the resulting diagram $\require{AMScd}$ $$\begin{CD} R' @>>> R\times_X Y@>>> Y\\ @VVV @VVV @VVV\\ Z @>>> R@>>> X \end{CD}$$ we get the left vertical arrow again by (ND) from $R' \to R$. However, as $X$ is excellent, the map $Y \to X$ is finite, hence $R' \to Z$ is a finite birational map to a normal scheme, so an isomorphism. This gives a lift $Z \to R \times_X Y$ in the above diagram. Any other such lift also induces a section of $R' \to Z$ by (ND) which implies uniqueness.