# The Stacks Project

## Tag 0C6H

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 28993–29000 (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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