Is this morphism the normalization of P^1 in this curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:32:30Z http://mathoverflow.net/feeds/question/71490 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71490/is-this-morphism-the-normalization-of-p1-in-this-curve Is this morphism the normalization of P^1 in this curve Homalor 2011-07-28T14:08:55Z 2011-07-28T15:11:17Z <p>Let $S$ be an integral Dedekind scheme.</p> <p>Let $f:X\longrightarrow \mathbf{P}^1_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme. </p> <p>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 $K(S)$.</p> <p><strong>Question.</strong> Is $f$ the normalization of $\mathbf{P}^1_S$ in the function field of $X_\eta$?</p> http://mathoverflow.net/questions/71490/is-this-morphism-the-normalization-of-p1-in-this-curve/71495#71495 Answer by Jason Starr for Is this morphism the normalization of P^1 in this curve Jason Starr 2011-07-28T14:31:48Z 2011-07-28T14:31:48Z <p>Yes. This follows from Zariski's Main Theorem (although there are probably more direct arguments in this case).</p> http://mathoverflow.net/questions/71490/is-this-morphism-the-normalization-of-p1-in-this-curve/71499#71499 Answer by Torsten Wedhorn for Is this morphism the normalization of P^1 in this curve Torsten Wedhorn 2011-07-28T15:11:17Z 2011-07-28T15:11:17Z <p>In your case the function field of <code>$X_{\eta}$</code> is the same as the function field of <code>$X$</code>. Thus the following general remark answers your question affirmatively.</p> <p>Assume $Y$ is an integral scheme and $L$ is an algebraic extension of the function field <code>$K(Y)$</code> of <code>$Y$</code>. Let <code>$\pi\colon X \to Y$</code> be an integral morphism of schemes such that <code>$X$</code> is integral and normal and such that $\pi$ induces on function fields the extension <code>$K(Y) \subset L = K(X)$</code>. 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.</p>