The Stacks project

Lemma 15.115.4. Let $A \to B \to C$ be extensions of discrete valuation rings with fraction fields $K \subset L \subset M$. Let $K_1/K$ be a finite extension.

  1. If $K_1$ is a (weak) solution for $A \to C$, then $K_1$ is a (weak) solution for $A \to B$.

  2. If $K_1$ is a (weak) solution for $A \to B$ and $L_1 = (L \otimes _ K K_1)_{red}$ is a product of fields which are (weak) solutions for $B \to C$, then $K_1$ is a (weak) solution for $A \to C$.

Proof. Let $L_1 = (L \otimes _ K K_1)_{red}$ and $M_1 = (M \otimes _ K K_1)_{red}$ and let $B_1 \subset L_1$ and $C_1 \subset M_1$ be the integral closure of $B$ and $C$. Note that $M_1 = (M \otimes _ L L_1)_{red}$ and that $L_1$ is a (nonempty) finite product of finite extensions of $L$. Hence the ring map $B_1 \to C_1$ is a finite product of ring maps of the form discussed in Remark 15.114.1. In particular, the map $\mathop{\mathrm{Spec}}(C_1) \to \mathop{\mathrm{Spec}}(B_1)$ is surjective. Choose a maximal ideal $\mathfrak m \subset C_1$ and consider the extensions of discrete valuation rings

\[ (A_1)_{A_1 \cap \mathfrak m} \to (B_1)_{B_1 \cap \mathfrak m} \to (C_1)_\mathfrak m \]

If the composition is weakly unramified, so is the map $(A_1)_{A_1 \cap \mathfrak m} \to (B_1)_{B_1 \cap \mathfrak m}$. If the residue field extension $\kappa _{A_1 \cap \mathfrak m} \to \kappa _\mathfrak m$ is separable, so is the subextension $\kappa _{A_1 \cap \mathfrak m} \to \kappa _{B_1 \cap \mathfrak m}$. Taking into account Lemma 15.111.5 this proves (1). A similar argument works for (2). $\square$

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Your email address will not be published.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 09ES. Beware of the difference between the letter 'O' and the digit '0'.