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Add internal references
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aisejohan committed Apr 19, 2024
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11 changes: 6 additions & 5 deletions homology.tex
Expand Up @@ -1185,9 +1185,9 @@ \section{Extensions}
as follows:
\begin{enumerate}
\item Given a morphism $B' \to B$ and an extension
$E$ of $B$ by $A$ we define $E' = E \times_B B'$
so that we have the following commutative diagram
of short exact sequences
$E$ of $B$ by $A$ we define $E' = E \times_B B'$.
By Lemma \ref{lemma-cartesian-kernel} we have the following
commutative diagram of short exact sequences
$$
\xymatrix{
0 \ar[r] & A \ar[r] \ar[d] & E' \ar[r] \ar[d] & B' \ar[r] \ar[d] & 0 \\
Expand All @@ -1197,8 +1197,9 @@ \section{Extensions}
The extension $E'$ is called the {\it pullback of $E$ via
$B' \to B$}.
\item Given a morphism $A \to A'$ and an extension
$E$ of $B$ by $A$ we define $E' = A' \amalg_A E$
so that we have the following commutative diagram
$E$ of $B$ by $A$ we define $E' = A' \amalg_A E$.
By Lemma \ref{lemma-cartesian-cocartesian}
we have the following commutative diagram
of short exact sequences
$$
\xymatrix{
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