# The Stacks Project

## Tag 010J

Definition 12.6.1. Let $\mathcal{A}$ be an abelian category. Let $A, B \in \mathop{\rm Ob}\nolimits(\mathcal{A})$. An extension $E$ of $B$ by $A$ is a short exact sequence $$0 \to A \to E \to B \to 0.$$ An morphism of extensions between two extensions $0 \to A \to E \to B \to 0$ and $0 \to A \to F \to B \to 0$ means a morphism $f : E \to F$ in $\mathcal{A}$ making the diagram $$\xymatrix{ 0 \ar[r] & A \ar[r] \ar[d]^{\text{id}} & E \ar[r] \ar[d]^f & B \ar[r] \ar[d]^{\text{id}} & 0 \\ 0 \ar[r] & A \ar[r] & F \ar[r] & B \ar[r] & 0 }$$ commutative. Thus, the extensions of $B$ by $A$ form a category.

The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 1092–1121 (see updates for more information).

\begin{definition}
\label{definition-extension}
Let $\mathcal{A}$ be an abelian category.
Let $A, B \in \Ob(\mathcal{A})$.
An {\it extension $E$ of $B$ by $A$} is a short
exact sequence
$$0 \to A \to E \to B \to 0.$$
An {\it morphism of extensions} between two
extensions $0 \to A \to E \to B \to 0$ and
$0 \to A \to F \to B \to 0$ means a morphism
$f : E \to F$ in $\mathcal{A}$ making the diagram
$$\xymatrix{ 0 \ar[r] & A \ar[r] \ar[d]^{\text{id}} & E \ar[r] \ar[d]^f & B \ar[r] \ar[d]^{\text{id}} & 0 \\ 0 \ar[r] & A \ar[r] & F \ar[r] & B \ar[r] & 0 }$$
commutative.
Thus, the extensions of $B$ by $A$ form a category.
\end{definition}

Comment #2376 by Fred Rohrer (site) on February 14, 2017 a 6:50 am UTC

Typo: $C$ should be $B$.

Comment #2432 by Johan (site) on February 17, 2017 a 3:05 pm UTC

Thanks, fixed here.

There are also 4 comments on Section 12.6: Homological Algebra.

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