I have a question that arises from following former thread of mine (look up in the comments to see the motivation):
Let $X$ be a scheme. It is well known that there exist a so called "normalization" $n: \overline{X} \to X$ where $\overline{X}$ is normal and has following universal property:
If $f: Y \to X$ is an arbitrary scheme morphism and $Y$ normal then $f$ factorizes through $\overline{X}$.
Now I'm curious if there exist a generalization of the "Normalization" procedure of a scheme in following sense:
Let $X$ be again a scheme with function field $K(X)$. Let $L/K(X)$ be an arbitrary (hope, that it's not too general) field extension of $K(X)$. Then there exist a morphism $m: W \to X$ with certain universal property blabla so that if $f: Y \to X$ with $Y$ has property blabla then $F$ factorize through $W$.
Does such construction exist? Sure, in case $L=K(X)$ this $W$ is exactly the normalization of $X$ but what about general extension of $K(X)$? Or is it hopeless?