The Stacks project

Lemma 15.111.4. Let $A$ be a valuation ring. Let $A \to B$ be an étale ring map and let $\mathfrak m \subset B$ be a prime lying over the maximal ideal of $A$. Then $A \subset B_\mathfrak m$ is an extension of valuation rings which is weakly unramified.

Proof. The ring $A$ has weak dimension $\leq 1$ by Lemma 15.95.18. Then $B$ has weak dimension $\leq 1$ by Lemmas 15.95.4 and 15.95.14. hence the local ring $B_\mathfrak m$ is a valuation ring by Lemma 15.95.18. Since the extension $A \subset B_\mathfrak m$ induces a finite extension of fraction fields, we see that the $\Gamma _ A$ has finite index in the value group of $B_{\mathfrak m}$. Thus for every $h \in B_\mathfrak m$ there exists an $n > 0$, an element $f \in A$, and a unit $w \in B_\mathfrak m$ such that $f = w h^ n$ in $B_\mathfrak m$. We will show that this implies $f = ug^ n$ for some $g \in A$ and unit $u \in A$; this will show that the value groups of $A$ and $B_\mathfrak m$ agree, as claimed in the lemma.

Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as the colimit of its local subrings which are essentially of finite type over $\mathbf{Z}$. Since $A$ is a normal domain (Algebra, Lemma 10.49.10), we may assume that each $A_ i$ is normal (here we use that taking normalizations the local rings remain essentially of finite type over $\mathbf{Z}$ by Algebra, Proposition 10.157.16). For some $i$ we can find an étale extension $A_ i \to B_ i$ such that $B = A \otimes _{A_ i} B_ i$, see Algebra, Lemma 10.142.3. Let $\mathfrak m_ i$ be the intersection of $B_ i$ with $\mathfrak m$. Then we may apply Lemma 15.111.3 to the ring map $A_ i \to (B_ i)_{\mathfrak m_ i}$ to conclude. The hypotheses of the lemma are satisfied because:

  1. $A_ i$ and $(B_ i)_{\mathfrak m_ i}$ are Noetherian as they are essentially of finite type over $\mathbf{Z}$,

  2. $A_ i \to (B_ i)_{\mathfrak m_ i}$ is flat as $A_ i \to B_ i$ is étale,

  3. $B_ i$ is normal as $A_ i \to B_ i$ is étale, see Algebra, Lemma 10.158.9,

  4. for every height $1$ prime of $A_ i$ there exists a height $1$ prime of $(B_ i)_{\mathfrak m_ i}$ lying over it by Algebra, Lemma 10.112.2 and the fact that $\mathop{\mathrm{Spec}}((B_ i)_{\mathfrak m_ i}) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective,

  5. the induced extensions $(A_ i)_\mathfrak p \to (B_ i)_\mathfrak q$ are unramified for every prime $\mathfrak q$ lying over a prime $\mathfrak p$ as $A_ i \to B_ i$ is étale.

This concludes the proof of the lemma. $\square$


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