Are hensel valuation rings N2?

Question

This seems like the kind of thing an expert should be able to answer off the top of their head:

Recall that a valuation ring is an integral domain $A$ such that for every $a \in Frac(A)$ we have either $a \in A$ or $a^{-1} \in A$. One of the many equivalent definitions of a Hensel valuation ring, is: a valuation ring $A$ is called Henselian if for every finite field extension $Frac(A) \subseteq L$, there exists a unique valuation ring $B \subseteq L$ such that $L = Frac(B)$ and $A = Frac(A) \cap B$.

Nowhere do I assume that the rings are Noetherian. I.e., the totally ordered set of prime ideals of a valuation ring is allowed to have any order type.

Let $A$ be a Hensel valuation ring and $Frac(A) \to L$ a finite field extension of its field of fractions. Let $B$ be the normalization of $A$ in $L$. A consequence of Chevalley's extension theorem is that $B$ is the valuation ring of the unique extension of the valuation of $Frac(A)$ to $L$ (see Corollary 3.1.4 in the Engler-Prestel book "Valued fields" for example).

Question: is $B$ a finite $A$-module in general?

Answer 1

The following counterexample is very similar to exercise 2 to Bourbaki's Algèbre commutative ch. VI §8 and works as well. Actually it is example 4.9 in Scholze's Perfectoid Spaces; there, the main point is that $B$ is an "almost finitely generated" $A$-module in the sense of "almost ring theory", so there is a way around the fact that it is not finitely generated. This fact, however, makes up our counterexample.

Let $p$ be an odd prime and for all $n \ge 1$, adjoin a $p^n$-th root of $p$, call it $\pi_n$, to the field of $p$-adic numbers $\mathbb{Q}_p$. The $p$-adic valuation uniquely extends to this field; take its completion with respect to this value. This is our field $K = Frac(A)$ with its local valuation ring $(A, \mathfrak{m})$. Note that the valuation $v: K^\times \rightarrow \mathbb{R}$ is of rank 1, but non-discrete; in fact, its value group is $\bigcup_{n \ge 1} \frac{1}{p^n} \mathbb{Z} = \mathbb{Z} [\frac{1}{p}]$. Because the rank is 1 and the field is complete, the valuation is Henselian (exercise 6b in Bourbaki, loc. cit.; first sentence of paragraph 4.1 in Engler-Prestel).

Now adjoin a square root of $p$ to get the quadratic field extension $L|K$, and extend $v$ to $L$. Convince yourself that for each $n \ge 0$, there is an element in $L$, call it $\rho_n$, which is a $2p^n$-th root of $p$ and thus is a primitive element for $L|K$. The value group of $L$ is $\frac{1}{2} \mathbb{Z}[\frac{1}{p}]$ and it is not hard to see that the valuation ring $B = \lbrace x \in L: v(x) \ge 0 \rbrace$ can be written as

$$B = A \oplus \bigcup_{n \ge 0} \rho_n A$$

as $A$-module. But because $v(\rho_n) = \frac{1}{2p^n} > \frac{1}{p^{n+1}} = v(\pi_{n+1})$ and $\pi_{n+1} \in \mathfrak{m}$, we have $\rho_n \in \mathfrak{m}B$ for all $n$, hence whole union on the right is contained in $\mathfrak{m}B$. So as $A$-modules,

$$B = A + \mathfrak{m}B .$$

Now if $B$ were finitely generated over $A$, Nakayama's lemma would imply $A = B$, which is absurd since $\rho_n \in B \setminus A$.

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Positive results: We have theorem 2 in Bourbaki's Algèbre commutative ch. VI §8 no. 5 which gives (in a more general setup, without the Henselian assumption) equivalent criteria for $B$ being a finitely generated $A$-module, especially the famous equality $[L:K] = \sum e_i f_i$. In Neukirch's Algebraic Number Theory ch. II §6, where the setup has a Henselian rank 1 valuation $v$, this equality is proven for $L|K$ separable and $v$ discrete; it is remarked that "both conditions are really necessary" but that for a complete field, one may drop the separability. Corresponding statements can be found in Serre's Local Fields ch. I §4 and ch. II §2, and also in Engler-Prestel, theorem 3.3.5. I guess that exercise 3 in Bourbaki loc. cit. is a counterexample with $v$ discrete and $L|K$ inseparable, I just do not see right away whether $v$ is Henselian there.

Answer 2

Exercise VI.8.3 from Bourbaki's Algèbre Commutative gives am example of a hensel discrete valuation ring and a finite (inseparable) extension of its fraction field for which the normalisation is not finite:

Define $k = \mathbb{F}_p(X_n)_{n \in \mathbb{N}}$.

Take $\alpha = \sum_{n = 0}^\infty X_n U^n \in k[[U]]$, although it looks like any transcendantal element $\alpha$ works, I don't see why we have to choose this one in particular.

Consider the sequence of fields $k(U) \subset k(U, \alpha^p) \subset k(U, \alpha) \subset k((U))$. The (complete) valuation ring $k[[U]] \subset k((U))$ induces valuation rings in all the others. Let $V_1 \subset V_2$ denote the two middle ones. We know the outer two are discrete valuation rings with residue field $k$ and therefore the same is true of the middle two.

The extension $k(U, \alpha) / k(U, \alpha^p)$ is finite and non-trivial, but induces isomorphisms of residue fields and value groups $(e = f = 1)$. It is generically purely inseparable, and therefore $R_2$ is the unique extension of $R_1$. Therefore $V_1 \subset V_2$ is not finite (Bourbaki AC Proposition 8 page 148).

Regardless of whether $V_1, V_2$ are henselian (its probably not too hard to show that they are not), we can take the associated hensel rings $V_1^h \subset V_2^h$, which are canonically valuation rings. Then $Frac(V_1^h) / Frac(V_1)$ is a separable algebraic (probably not finite) extension and therefore $Frac(V_1^h)$ does not contain $\alpha$. Henselisation preserves residue fields and value groups (Theorem 5.2.5 in Engler-Prestel) and so $V_1^h \subset V_2^h$ is not finite as long as $Frac(V_2^h) / Frac(V_1^h)$ is still purely inseparable of degree $p$.

For this we show notice that $Frac(V_2^h) = Frac(V_1^h) \otimes_{Frac(V_1)} Frac(V_2)$.