diff --git a/modules.tex b/modules.tex
index 59a13d62c..16d688fb1 100644
--- a/modules.tex
+++ b/modules.tex
@@ -890,7 +890,7 @@ \section{Modules of finite type}
 Choose an open neighbourhood $U \subset X$ such that $\mathcal{F}$ is
 generated by $s_1, \ldots, s_n \in \mathcal{F}(U)$ over $U$.
 By assumption of surjectivity of $\varphi_x$,
-after shrinking $V$ we may assume that $s_i = \varphi(t_i)$
+after shrinking $U$ we may assume that $s_i = \varphi(t_i)$
 for some $t_i \in \mathcal{G}(U)$.
 Then $U$ works.
 \end{proof}
diff --git a/schemes.tex b/schemes.tex
index cf45f5505..1b2f4366b 100644
--- a/schemes.tex
+++ b/schemes.tex
@@ -451,7 +451,7 @@ \section{Closed immersions of locally ringed spaces}
 \label{lemma-restrict-map-to-closed}
 Let $f : X \to Y$ be a morphism of locally ringed spaces.
 Let $\mathcal{I} \subset \mathcal{O}_Y$ be a sheaf of
-ideals which is locally generated by functions.
+ideals which is locally generated by sections.
 Let $i : Z \to Y$ be the closed subspace associated to the
 sheaf of ideals $\mathcal{I}$.
 Let $\mathcal{J}$ be the image of the map