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Tag 031F

Chapter 10: Commutative Algebra > Section 10.118: Around Krull-Akizuki

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 27738–27749 (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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