Dimension and qf maps in decent-spaces

decent-spaces.tex

@@ -2317,6 +2317,17 @@ \section{Points on spaces}
 This is Algebra, Lemma \ref{algebra-lemma-dimension-inequality-quasi-finite}.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-dimension-quasi-finite}
+Let $S$ be a scheme. Let $X \to Y$ be a locally quasi-finite morphism
+of algebraic spaces over $S$. Then $\dim(X) \leq \dim(Y)$.
+\end{lemma}
+
+\begin{proof}
+This follows from Lemma \ref{lemma-dimension-local-ring-quasi-finite}
+and Properties of Spaces, Lemma \ref{spaces-properties-lemma-dimension}.
+\end{proof}
+
 \noindent
 The following lemma is a tiny bit stronger than
 Properties of Spaces,