# The Stacks Project

## Tag 031F

Lemma 10.118.10. Let $R \to S$ be a homomorphism of domains inducing an injection of fraction fields $K \subset L$. If $R$ is Noetherian local of dimension $1$ and $[L : K] < \infty$ then

1. each prime ideal $\mathfrak n_i$ of $S$ lying over the maximal ideal $\mathfrak m$ of $R$ is maximal,
2. there are finitely many of these, and
3. $[\kappa(\mathfrak n_i) : \kappa(\mathfrak m)] < \infty$ for each $i$.

Proof. Pick $x \in \mathfrak m$ nonzero. Apply Lemma 10.118.9 to the submodule $S \subset L \cong K^{\oplus n}$ where $n = [L : K]$. Thus the ring $S/xS$ has finite length over $R$. It follows that $S/\mathfrak m S$ has finite length over $\kappa(\mathfrak m)$. In other words, $\dim_{\kappa(\mathfrak m)} S/\mathfrak m S$ is finite (Lemma 10.51.6). Thus $S/\mathfrak mS$ is Artinian (Lemma 10.52.2). The structural results on Artinian rings implies parts (1) and (2), see for example Lemma 10.52.6. Part (3) is implied by the finiteness established above. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 27781–27792 (see updates for more information).

\begin{lemma}
\label{lemma-finite-extension-residue-fields-dimension-1}
Let $R \to S$ be a homomorphism of domains inducing an
injection of fraction fields $K \subset L$. If $R$ is Noetherian
local of dimension $1$ and $[L : K] < \infty$ then
\begin{enumerate}
\item each prime ideal $\mathfrak n_i$ of $S$ lying over
the maximal ideal $\mathfrak m$ of $R$ is maximal,
\item there are finitely many of these, and
\item $[\kappa(\mathfrak n_i) : \kappa(\mathfrak m)] < \infty$ for each $i$.
\end{enumerate}
\end{lemma}

\begin{proof}
Pick $x \in \mathfrak m$ nonzero. Apply Lemma \ref{lemma-finite-length}
to the submodule $S \subset L \cong K^{\oplus n}$ where $n = [L : K]$.
Thus the ring $S/xS$ has finite length over $R$. It follows that
$S/\mathfrak m S$ has finite length over $\kappa(\mathfrak m)$.
In other words, $\dim_{\kappa(\mathfrak m)} S/\mathfrak m S$
is finite (Lemma \ref{lemma-dimension-is-length}). Thus $S/\mathfrak mS$
is Artinian (Lemma \ref{lemma-finite-dimensional-algebra}). The
structural results on Artinian rings implies parts (1) and (2), see
for example Lemma \ref{lemma-artinian-finite-length}.
Part (3) is implied by the finiteness established above.
\end{proof}

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