Let A be a Dedekind domain, K its field of fractions, L/K a finite extension, B the integral closure of A in L. By the Krull-Akizuki theorem, B is noetherian, hence B is a Dedekind domain. In the following two cases, B is an A-module of finite type.

1) L/K is separable.

2) A is an algebra of finite type over a field.

I'm trying to find an example where B is not an A-module of finite type. After spending some time, I gave up. It seems to be a very hard problem.

  • $\begingroup$ Theorem 100, Kaplansky, Commutative Rings. $\endgroup$ – user26857 Mar 2 '15 at 2:20

Look at $k(x)$ with $k$ a field of characteristic $p>0$. The extension $k((x))/k(x)$ is transcendental so let $\alpha\in k[[x]]$ be transcendental and let $\beta =\alpha^p$. Then $L:=k(x,\alpha )$ is a purely inseparable extension of $K:=k(x,\beta)$. (Classic.) The DVR $B=k[[x]]\cap L$ is the integral closure of the DVR $A=k[[x]]\cap K$ and $x$ is a prime element of $A$. Now each $n\in\mathbb{N}$ the element $\beta$ can be written as $$\beta=f_n^p + x^{pn}y_n , f_n\in k[x], y_n\in k[[x]].$$ (Look at $\alpha =c_0+c_1x+c_2x^2+c_3x^3+\ldots$ so that $\beta =c_0^p+c_1^px^p+c_2^px^{2p}+c_3^px^{3p}+\ldots$.)

Then $y_n\in A$ and you get $$y_n^{1/p} =x^{-n}(-f_n+\alpha )\;\; (Eq).$$ Now if $B$ is a finite $A$-module you can find $d\in A\backslash \{0\}$ such that $$dB\subseteq A+A\alpha +A\alpha^2 +\ldots +A\alpha^{p-1} :=R.$$ Then $d y_n^{1/p}\in R$, which using (Eq) gives that $d$ is divisible by $x^n$ for every $n$. (Note that $1, \alpha ,\ldots ,\alpha^{p-1}$ are linearly independent over $K$.)

  • $\begingroup$ This is in Zariski's book on valuations. $\endgroup$ – ujsgeyrr1f0d0d0r0h1h0j0j_juj Mar 2 '15 at 2:27
  • $\begingroup$ I was referring to Zariski's and Samuel's book "Commutative algebra". Actually I checked out but this is not in this book. My bad. Speaking of the example I gave, I learned it in school long tim ago, but I remember that it was quoted as a example dued to Zariski. Strange. $\endgroup$ – ujsgeyrr1f0d0d0r0h1h0j0j_juj Mar 2 '15 at 11:30

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