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Tag 0C6H

Chapter 10: Commutative Algebra > Section 10.121: Quasi-finite maps

Lemma 10.121.9. Let $A \to B$ and $B \to C$ be finite type ring homomorphisms. Let $\mathfrak r$ be a prime of $C$ lying over $\mathfrak q \subset B$ and $\mathfrak p \subset A$. If $A \to C$ is quasi-finite at $\mathfrak r$, then $B \to C$ is quasi-finite at $\mathfrak r$.

Proof. Using property (3) of Lemma 10.121.2: By assumption there exists some $c \in C$ such that $$ \{\mathfrak r' \subset C \text{ lying over }\mathfrak p\} \cap D(c) = \{\mathfrak{r}\}. $$ Since the primes $\mathfrak r' \subset C$ lying over $\mathfrak q$ form a subset of the primes $\mathfrak r' \subset C$ lying over $\mathfrak p$ we conclude. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 29047–29054 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-finite-permanence}
    Let $A \to B$ and $B \to C$ be finite type ring homomorphisms.
    Let $\mathfrak r$ be a prime of $C$ lying over
    $\mathfrak q \subset B$ and $\mathfrak p \subset A$.
    If $A \to C$ is quasi-finite at $\mathfrak r$, then
    $B \to C$ is quasi-finite at $\mathfrak r$.
    \end{lemma}
    
    \begin{proof}
    Using property (3) of Lemma \ref{lemma-isolated-point-fibre}:
    By assumption there exists some $c \in C$ such that
    $$
    \{\mathfrak r' \subset C \text{ lying over }\mathfrak p\} \cap D(c) =
    \{\mathfrak{r}\}.
    $$
    Since the primes $\mathfrak r' \subset C$ lying over $\mathfrak q$
    form a subset of the primes $\mathfrak r' \subset C$ lying over
    $\mathfrak p$ we conclude.
    \end{proof}

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