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15.97 Extensions of discrete valuation rings

In this section and the next few we use the following definitions.

Definition 15.97.1. We say that $A \to B$ or $A \subset B$ is an extension of discrete valuation rings if $A$ and $B$ are discrete valuation rings and $A \to B$ is injective and local. In particular, if $\pi _ A$ and $\pi _ B$ are uniformizers of $A$ and $B$, then $\pi _ A = u \pi _ B^ e$ for some $e \geq 1$ and unit $u$ of $B$. The integer $e$ does not depend on the choice of the uniformizers as it is also the unique integer $\geq 1$ such that

\[ \mathfrak m_ A B = \mathfrak m_ B^ e \]

The integer $e$ is called the ramification index of $B$ over $A$. We say that $B$ is weakly unramified over $A$ if $e = 1$. If the extension of residue fields $\kappa _ A = A/\mathfrak m_ A \subset \kappa _ B = B/\mathfrak m_ B$ is finite, then we set $f = [\kappa _ B : \kappa _ A]$ and we call it the residual degree or residue degree of the extension $A \subset B$.

Note that we do not require the extension of fraction fields to be finite.

Lemma 15.97.2. Let $A \subset B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. If the extension $L/K$ is finite, then the residue field extension is finite and we have $ef \leq [L : K]$.

Proof. Finiteness of the residue field extension is Algebra, Lemma 10.118.10. The inequality follows from Algebra, Lemmas 10.118.9 and 10.51.12. $\square$

Lemma 15.97.3. Let $A \subset B \subset C$ be extensions of discrete valuation rings. Then the ramification indices of $B/A$ and $C/B$ multiply to give the ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$. Similarly for the residual degrees in case they are finite.

Proof. This is immediate from the definitions and Fields, Lemma 9.7.7. $\square$

Lemma 15.97.4. Let $A \subset B$ be an extension of discrete valuation rings inducing the field extension $K \subset L$. If the characteristic of $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then the ramification index $e$ is a power of $p$.

Proof. Write $\pi _ A = u \pi _ B^ e$ for some $u \in B^*$. On the other hand, we have $\pi _ B^ q \in K$ for some $p$-power $q$. Write $\pi _ B^ q = v \pi _ A^ k$ for some $v \in A^*$ and $k \in \mathbf{Z}$. Then $\pi _ A^ q = u^ q \pi _ B^{qe} = u^ q v^ e \pi _ A^{ke}$. Taking valuations in $B$ we conclude that $ke = q$. $\square$

In the following lemma we discuss what it means for an extension $A \subset B$ of discrete valuation rings to be “unramified”, i.e., have ramification index $1$ and separable (possibly nonalgebraic) extension of residue fields. However, we cannot use the term “unramified” itself because there already exists a notion of an unramified ring map, see Algebra, Section 10.147.

Lemma 15.97.5. Let $A \subset B$ be an extension of discrete valuation rings. The following are equivalent

  1. $A \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology, and

  2. $A \to B$ is weakly unramified and $\kappa _ A \subset \kappa _ B$ is a separable field extension.

Proof. This follows from Proposition 15.39.5 and Algebra, Proposition 10.152.9. $\square$

Remark 15.97.6. Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite separable field extension. Let $B \subset L$ be the integral closure of $A$ in $L$. Picture:

\[ \xymatrix{ B \ar[r] & L \\ A \ar[u] \ar[r] & K \ar[u] } \]

By Algebra, Lemma 10.155.8 the ring extension $A \subset B$ is finite, hence $B$ is Noetherian. By Algebra, Lemma 10.111.4 the dimension of $B$ is $1$, hence $B$ is a Dedekind domain, see Algebra, Lemma 10.119.17. Let $\mathfrak m_1, \ldots , \mathfrak m_ n$ be the maximal ideals of $B$ (i.e., the primes lying over $\mathfrak m_ A$). We obtain extensions of discrete valuation rings

\[ A \subset B_{\mathfrak m_ i} \]

and hence ramification indices $e_ i$ and residue degrees $f_ i$. We have

\[ [L : K] = \sum \nolimits _{i = 1, \ldots , n} e_ i f_ i \]

by Algebra, Lemma 10.120.8 applied to a uniformizer in $A$. We observe that $n = 1$ if $A$ is henselian (by Algebra, Lemma 10.148.4), e.g. if $A$ is complete.

Definition 15.97.7. Let $A$ be a discrete valuation ring with fraction field $K$. Let $L \supset K$ be a finite separable extension. With $B$ and $\mathfrak m_ i$, $i = 1, \ldots , n$ as in Remark 15.97.6 we say the extension $L/K$ is

  1. unramified with respect to $A$ if $e_ i = 1$ and the extension $\kappa _ A \subset \kappa (\mathfrak m_ i)$ is separable for all $i$,

  2. tamely ramified with respect to $A$ if either the characteristic of $\kappa _ A$ is $0$ or the characteristic of $\kappa _ A$ is $p > 0$, the field extensions $\kappa _ A \subset \kappa (\mathfrak m_ i)$ are separable, and the ramification indices $e_ i$ are prime to $p$, and

  3. totally ramified with respect to $A$ if $n = 1$ and the residue field extension $\kappa _ A \subset \kappa (\mathfrak m_1)$ is trivial.

If the discrete valuation ring $A$ is clear from context, then we sometimes say $L/K$ is unramified, totally ramified, or tamely ramified for short.

For unramified extensions we have the following basic lemma.

Lemma 15.97.8. Let $A$ be a discrete valuation ring with fraction field $K$.

  1. If $M/L/K$ are finite separable extensions and $M$ is unramified with respect to $A$, then $L$ is unramified with respect to $A$.

  2. If $L/K$ is a finite separable extension which is unramified with respect to $A$, then there exists a Galois extension $M/K$ containing $L$ which is unramified with respect to $A$.

  3. If $L_1/K$, $L_2/K$ are finite separable extensions which are unramified with respect to $A$, then there exists a a finite separable extension $L/K$ which is unramified with respect to $A$ containing $L_1$ and $L_2$.

Proof. We will use the results of the discussion in Remark 15.97.6 without further mention.

Proof of (1). Let $C/B/A$ be the integral closures of $A$ in $M/L/K$. Since $C$ is a finite ring extension of $B$, we see that $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(B)$ is surjective. Hence for ever maximal ideal $\mathfrak m \subset B$ there is a maximal ideal $\mathfrak m' \subset C$ lying over $\mathfrak m$. By the multiplicativity of ramification indices (Lemma 15.97.3) and the assumption, we conclude that the ramification index of $B_\mathfrak m$ over $A$ is $1$. Since $\kappa (\mathfrak m')/\kappa _ A$ is finite separable, the same is true for $\kappa (\mathfrak m)/\kappa _ A$.

Proof of (2). Let $M$ be the normal closure of $L$ over $K$, see Fields, Definition 9.16.4. Then $M/K$ is Galois by Fields, Lemma 9.21.5. On the other hand, there is a surjection

\[ L \otimes _ K \ldots \otimes _ K L \longrightarrow M \]

of $K$-algebras, see Fields, Lemma 9.16.6. Let $B$ be the integral closure of $A$ in $L$ as in Remark 15.97.6. The condition that $L$ is unramified with respect to $A$ exactly means that $A \to B$ is an étale ring map, see Algebra, Lemma 10.141.7. By permanence properties of étale ring maps we see that

\[ B \otimes _ A \ldots \otimes _ A B \]

is étale over $A$, see Algebra, Lemma 10.141.3. Hence the displayed ring is a product of Dedekind domains, see Lemma 15.43.4. We conclude that $M$ is the fraction field of a Dedekind domain finite étale over $A$. This means that $M$ is unramified with respect to $A$ as desired.

Proof of (3). Let $B_ i \subset L_ i$ be the integral closure of $A$. Argue in the same manner as above to show that $B_1 \otimes _ A B_2$ is finite étale over $A$. Details omitted. $\square$

Lemma 15.97.9. Let $A$ be a discrete valuation ring with fraction field $K$. Let $M/L/K$ be finite separable extensions. Let $B$ be the integral closure of $A$ in $L$. If $L/K$ is unramified with respect to $A$ and $M/L$ is unramified with respect to $B_\mathfrak m$ for every maximal ideal $\mathfrak m$ of $B$, then $M/K$ is unramified with respect to $A$.

Proof. Let $C$ be the integral closure of $A$ in $M$. Every maximal ideal $\mathfrak m'$ of $C$ lies over a maximal ideal $\mathfrak m$ of $B$. Then the lemma follows from the multiplicativity of ramification indices (Lemma 15.97.3) and the fact that we have the tower $\kappa (\mathfrak m')/\kappa (\mathfrak m)/\kappa _ A$ of finite extensions of fields. $\square$


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