Is this morphism the normalization of P^1 in this curve - MathOverflow

Is this morphism the normalization of P^1 in this curve

2011-07-28T14:08:55Z

Let $S$ be an integral Dedekind scheme.

Let $f:X\longrightarrow \mathbf{P}^1_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme.

Let $\eta$ be the generic point of $S$. Note that $f_\eta:X_\eta\longrightarrow \mathbf{P}^1_{K(S)}$ is a finite morphism of curves over $K(S)$.

Question. Is $f$ the normalization of $\mathbf{P}^1_S$ in the function field of $X_\eta$?

Answer by Jason Starr for Is this morphism the normalization of P^1 in this curve

2011-07-28T14:31:48Z

Yes. This follows from Zariski's Main Theorem (although there are probably more direct arguments in this case).

Answer by Torsten Wedhorn for Is this morphism the normalization of P^1 in this curve

2011-07-28T15:11:17Z

In your case the function field of $X_{\eta}$ is the same as the function field of $X$ . Thus the following general remark answers your question affirmatively.

Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field $K(Y)$ of $Y$ . Let $\pi\colon X \to Y$ be an integral morphism of schemes such that $X$ is integral and normal and such that $\pi$ induces on function fields the extension $K(Y) \subset L = K(X)$ . Then $X$ is the normalization of $Y$ in $L$. In fact this follows essentially from the definition of "normalization" and the fact that integral ring homomorphisms are stable under localization.