Lemma 15.115.9. Let $A \subset B \subset C$ be extensions of discrete valuation rings with fractions fields $K \subset L \subset M$. Let $\pi \in A$ be a uniformizer. Assume

1. $B$ is a Nagata ring,

2. $A \subset B$ is weakly unramified,

3. $M$ is a degree $p$ purely inseparable extension of $L$.

Then either

1. $A \to C$ is weakly unramified, or

2. $C = B[\pi ^{1/p}]$, or

3. there exists a degree $p$ separable extension $K_1/K$ totally ramified with respect to $A$ such that $L_1 = L \otimes _ K K_1$ and $M_1 = M \otimes _ K K_1$ are fields and the maps of integral closures $A_1 \to B_1 \to C_1$ are weakly unramified extensions of discrete valuation rings.

Proof. Let $e$ be the ramification index of $C$ over $B$. If $e = 1$, then we are done. If not, then $e = p$ by Lemmas 15.111.2 and 15.111.4. This in turn implies that the residue fields of $B$ and $C$ agree. Choose a uniformizer $\pi _ C$ of $C$. Write $\pi _ C^ p = u \pi$ for some unit $u$ of $C$. Since $\pi _ C^ p \in L$, we see that $u \in B^*$. Also $M = L[\pi _ C]$.

Suppose there exists an integer $m \geq 0$ such that

$u = \sum \nolimits _{0 \leq i < m} b_ i^ p \pi ^ i + b \pi ^ m$

with $b_ i \in B$ and with $b \in B$ an element whose image in $\kappa _ B$ is not a $p$th power. Choose an extension $K_1/K$ as in Lemma 15.115.7 with $n = m + 2$ and denote $\pi '$ the uniformizer of the integral closure $A_1$ of $A$ in $K_1$ such that $\pi = (\pi ')^ p + (\pi ')^{np} a$ for some $a \in A_1$. Let $B_1$ be the integral closure of $B$ in $L \otimes _ K K_1$. Observe that $A_1 \to B_1$ is weakly unramified by Lemma 15.115.3. In $B_1$ we have

$u \pi = \left(\sum \nolimits _{0 \leq i < m} b_ i (\pi ')^{i + 1}\right)^ p + b (\pi ')^{(m + 1)p} + (\pi ')^{np} b_1$

for some $b_1 \in B_1$ (computation omitted). We conclude that $M_1$ is obtained from $L_1$ by adjoining a $p$th root of

$b + (\pi ')^{n - m - 1} b_1$

Since the residue field of $B_1$ equals the residue field of $B$ we see from Lemma 15.115.8 that $M_1/L_1$ has degree $p$ and the integral closure $C_1$ of $B_1$ is weakly unramified over $B_1$. Thus we conclude in this case.

If there does not exist an integer $m$ as in the preceding paragraph, then $u$ is a $p$th power in the $\pi$-adic completion of $B_1$. Since $B$ is Nagata, this means that $u$ is a $p$th power in $B_1$ by Algebra, Lemma 10.162.18. Whence the second case of the statement of the lemma holds. $\square$

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