Lemma 15.111.9. 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 15.111.3) 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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