# Finiteness of normalization of Noetherian normal domain

I have the following question:

Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is it true then that $B$ is finite over $A$?

If $L/K$ is separable, it is true (there is a usual proof with considering the non-degenerate bilinear form $tr(xy)$). What about a non-separable extension?

Thanks, Sasha

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–  ACL Nov 8 '11 at 10:31
Depends on your definition of "geometry". Discrete valuation rings of equal characteristic $p$ don't have to be excellent, and if you do analytic geometry in positive characteristic you can encounter them. –  Alex Nov 9 '11 at 19:25