Proposition 15.105.21. Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If $B$ is essentially of finite type over $A$, then there exists a finite extension $K \subset K_1$ which is a solution for $A \to B$ as defined in Definition 15.105.1.

See [Lemma 2.13, alterations] for a special case.

**Proof.**
Observe that a weak solution is a solution if the residue field of $A$ is perfect, see Lemma 15.101.5. Thus the proposition follows immediately from Theorem 15.105.19 if the residue characteristic of $A$ is $0$ (and in fact we do not need the assumption that $A \to B$ is essentially of finite type). If the residue characteristic of $A$ is $p > 0$ we will also deduce it from Epp's theorem.

Let $x_ i \in A$, $i \in I$ be a set of elements mapping to a $p$-base of the residue field $\kappa $ of $A$. Set

where the transition maps send $t_{i, n + 1}$ to $t_{i, n}^ p$. Observe that $A'$ is a filtered colimit of weakly unramified finite extensions of discrete valuation rings over $A$. Thus $A'$ is a discrete valuation ring and $A \to A'$ is weakly unramified. By construction the residue field $\kappa ' = A'/\mathfrak m_ A A'$ is the perfection of $\kappa $.

Let $K'$ be the fraction field of $A'$. We may apply Lemma 15.105.20 to the extension $K \subset K'$. Thus $B'$ is a finite product of Dedekind domains. Let $\mathfrak m_1, \ldots , \mathfrak m_ n$ be the maximal ideals of $B'$. Using Epp's theorem (Theorem 15.105.19) we find a weak solution $K' \subset K'_ i$ for each of the extensions $A' \subset B'_{\mathfrak m_ i}$. Since the residue field of $A'$ is perfect, these are actually solutions. Let $K' \subset K'_1$ be a finite extension which contains each $K'_ i$. Then $K' \subset K'_1$ is still a solution for each $A' \subset B'_{\mathfrak m_ i}$ by Lemma 15.104.3.

Let $A'_1$ be the integral closure of $A$ in $K'_1$. Note that $A'_1$ is a Dedekind domain by the discussion in Remark 15.104.1 applied to $K' \subset K'_1$. Thus Lemma 15.105.20 applies to $K \subset K'_1$. Therefore the integral closure $B'_1$ of $B$ in $L'_1 = (L \otimes _ K K'_1)_{red}$ is a Dedekind domain and because $K' \subset K'_1$ is a solution for each $A' \subset B'_{\mathfrak m_ i}$ we see that $(A'_1)_{A'_1 \cap \mathfrak m} \to (B'_1)_{\mathfrak m}$ is formally smooth in the $\mathfrak m$-adic topology for each maximal ideal $\mathfrak m \subset B'_1$.

By construction, the field $K'_1$ is a filtered colimit of finite extensions of $K$. Say $K'_1 = \mathop{\mathrm{colim}}\nolimits _{i \in I} K_ i$. For each $i$ let $A_ i$, resp. $B_ i$ be the integral closure of $A$, resp. $B$ in $K_ i$, resp. $L_ i = (L \otimes _ K K_ i)_{red}$. Then it is clear that

Since the ring maps $A_ i \to A'_1$ and $B_ i \to B'_1$ are injective integral ring maps and since $A'_1$ and $B'_1$ have finite spectra, we see that for all $i$ large enough the ring maps $A_ i \to A'_1$ and $B_ i \to B'_1$ are bijective on spectra. Once this is true, for all $i$ large enough the maps $A_ i \to A'_1$ and $B_ i \to B'_1$ will be weakly unramified (once the uniformizer is in the image). It follows from multiplicativity of ramification indices that $A_ i \to B_ i$ induces weakly unramified maps on all localizations at maximal ideals of $B_ i$ for such $i$. Increasing $i$ a bit more we see that

induces surjective maps on residue fields (because the residue fields of $B'_1$ are finitely generated over those of $A'_1$ by Lemma 15.105.20). Picture of residue fields at maximal ideals lying under a chosen maximal ideal of $B'_1$:

Thus $\kappa _{B_ i}$ is a finitely generated extension of $\kappa _{A_ i}$ such that the compositum of $\kappa _{B_ i}$ and $\kappa _{A'_1}$ in $\kappa _{B'_1}$ is separable over $\kappa _{A'_1}$. Then that happens already at a finite stage: for example, say $\kappa _{B'_1}$ is finite separable over $\kappa _{A'_1}(x_1, \ldots , x_ n)$, then just increase $i$ such that $x_1, \ldots , x_ n$ are in $\kappa _{B_ i}$ and such that all generators satisfy separable polynomial equations over $\kappa _{A_ i}(x_1, \ldots , x_ n)$. This means that $A_ i \to (B_ i)_\mathfrak m$ is formally smooth in the $\mathfrak m$-adic topology for all maximal ideals $\mathfrak m$ of $B_ i$ and the proof is complete. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: