# stacks/stacks-project

Strengthen a lemma

Thanks to Manuel Hoff
https://stacks.math.columbia.edu/tag/0C6H#comment-3952
 \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 Let $A \to B$ and $B \to C$ be ring homomorphisms such that $A \to C$ is of finite type. 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 Observe that $B \to C$ is of finite type (Lemma \ref{lemma-compose-finite-type}) so that the statement makes sense. Let us use characterization (3) of Lemma \ref{lemma-isolated-point-fibre}. If $A \to C$ is quasi-finite at $\mathfrak r$, then 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. $\mathfrak p$ we conclude $B \to C$ is quasi-finite at $\mathfrak r$. \end{proof} \noindent