The Stacks project

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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 15.115: Eliminating ramification

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09EY. Beware of the difference between the letter 'O' and the digit '0'.