Minor edits

Signed-off-by: zeramorphic <[email protected]>

iii/commalg/02_tensor_products.tex

@@ -626,10 +626,16 @@ \subsection{Restriction and extension of scalars}
     Then
     \[ M \otimes_R N \simeq M \otimes_S (S \otimes_R N) \]
     as \( S \)-modules, where
-    \[ m \otimes n \mapsto m \otimes (1 \otimes n);\quad m \otimes (s \otimes n) \mapsto sm \otimes n \]
+    \[ m \otimes n \mapsto m \otimes (1 \otimes n);\quad sm \otimes n \mapsfrom m \otimes (s \otimes n) \]
 \end{proposition}
 \begin{proof}
-    %TODO/ES1
+    The map \( (m, n) \mapsto m \otimes (1 \otimes n) \) is \( R \)-bilinear, so the map \( f \) mapping \( m \otimes n \) to \( m \otimes (1 \otimes n) \) is well-defined as a map of \( R \)-modules.
+    We show it is \( S \)-linear on pure tensors.
+    \[ f(s(m \otimes n)) = f(sm \otimes n) = sm \otimes (1 \otimes n) = s (m \otimes (1 \otimes n)) = s f(m \otimes n) \]
+    For a fixed \( m \in M \), the map \( s \otimes n \mapsto sm \otimes n \) is well-defined and \( S \)-linear.
+    This collection of maps is \( S \)-linear in its parameter \( m \), so we obtain an \( S \)-bilinear map \( (m, s \otimes n) \mapsto sm \otimes n \).
+    Hence, we obtain a map \( g \) mapping \( m \otimes (s \otimes n) \) to \( sm \otimes n \), as desired.
+    One can easily check that \( f \) and \( g \) are inverses on pure tensors.
 \end{proof}
 \begin{proposition}
     Let \( M, M' \) be \( S \)-modules and \( N, N' \) be \( R \)-modules.
@@ -871,7 +877,8 @@ \subsection{Exactness properties of the tensor product}
 \end{proof}
 By the universal property of the tensor product,
 \[ \Hom_R(M \otimes_R N, L) \simeq \operatorname{Bilin}_R(M \times N, L) \simeq \Hom_R(N, \Hom_R(M, L)) \]
-mapping \( \varphi \mapsto n \mapsto m \mapsto \varphi(m \otimes n) \) and \( \varphi \mapsto m \otimes n \mapsto \varphi(m)(n) \).
+given by
+\[ \varphi \mapsto (n \mapsto m \mapsto \varphi(m \otimes n)) ;\quad (m \otimes n \mapsto \varphi(m)(n)) \mapsfrom \varphi \]
 This bijection is \emph{natural}, in the sense that many commutative diagrams involving them will commute.
 \begin{proposition}
     Let \( M \) be an \( R \)-module.

iii/commalg/03_localisation.tex

@@ -2,7 +2,7 @@ \subsection{Definitions}
 \begin{definition}
     A \emph{multiplicative set} or \emph{multiplicatively closed set} \( S \subseteq R \) is a subset such that \( 1 \in S \) and if \( a, b \in S \), then \( ab \in S \).
     If \( U \subseteq R \) is any set, its \emph{multiplicative closure} \( S \) of \( U \) is the set
-    \[ \qty{\prod_{i = 1}^n u_i \mid n \geq 0, u_i \in U} \]
+    \[ \qty{\prod_{i = 1}^n u_i \midd n \geq 0, u_i \in U} \]
     which is the smallest multiplicatively closed set containing \( U \).
 \end{definition}
 \begin{example}