Let $R$ be a DVR, $A$ a finitely generated integral $R$-algebra, and $A'$ the normalization of $A$ in the fraction field of $A$. Then is $A'$ finite as an $A$-module?
I know that if $R$ is a field, then it's true. And I know this is true, see here, section 2 in page 8. But I don't know its proof. So please show it or suggest some references.
Thank you.
The property you state is related to the property of $R$ being excellent, and more specifically to weaker condition of whether $R$ has the N-2 property or the stronger Nagata property; see [Matsumura, §31] and [Illusie–Laszlo–Orgogozo, Exp. I].
Not all DVR's satisfy your property. We give an example below, due to Nagata. On the other hand, both Dedekind domains of characteristic zero and complete local rings are excellent [Illusie–Laszlo–Orgogozo, Exp. I, Prop. 3.1 and §4], hence satisfy your property.
Example [Nagata, App. A1, Ex. 3]. We give an example of a DVR $R$ and an algebra of finite type $A$ over $R$ such that the normalization $A'$ of $A$ in $\operatorname{Frac}(A)$ is not finite over $A$. Let $k$ be a field of characteristic $p > 0$ such that $[k : k^p] = \infty$, and consider the ring $$R = \biggl\{\sum_{i=0}^\infty a_ix^i \in k[[x]] \biggm\vert [k^p(a_0,a_1,\ldots):k^p] < \infty \biggr\}.$$ This is a DVR by [Nagata, App. A1, (E3.1)]. Let $\{b_1,b_2,\ldots\} \subseteq k$ be a sequence of elements in $k$ that are $p$-independent over $k^p$. Set $$c = \sum_{i=0}^\infty b_ix^i,$$ and consider the ring $A = R[c]$. Then, the normalization $A'$ of $A$ in $\operatorname{Frac}(A)$ is not finite over $A$ by [Nagata, App. A1, (E3.2)].