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Tag 010I

12.6. Extensions

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 $$ 0 \to A \to E \to B \to 0. $$

By abuse of language we often omit mention of the morphisms $A \to E$ and $E \to B$, although they are definitively part of the structure of an extension.

Definition 12.6.2. Let $\mathcal{A}$ be an abelian category. Let $A, B \in \mathop{\rm Ob}\nolimits(\mathcal{A})$. The set of isomorphism classes of extensions of $B$ by $A$ is denoted $$ \mathop{\rm Ext}\nolimits_\mathcal{A}(B, A). $$ This is called the $\mathop{\rm Ext}\nolimits$-group.

This definition works, because by our conventions $\mathcal{A}$ is a set, and hence $\mathop{\rm Ext}\nolimits_\mathcal{A}(B, A)$ is a set. In any of the cases of ''big'' abelian categories listed in Categories, Remark 4.2.2. one can check by hand that $\mathop{\rm Ext}\nolimits_\mathcal{A}(B, A)$ is a set as well. Also, we will see later that this is always the case when $\mathcal{A}$ has either enough projectives or enough injectives. Insert future reference here.

Actually we can turn $\mathop{\rm Ext}\nolimits_\mathcal{A}(-, -)$ into a functor $$ \mathcal{A} \times \mathcal{A}^{opp} \longrightarrow \textit{Sets}, \quad (A, B) \longmapsto \mathop{\rm Ext}\nolimits_\mathcal{A}(B, A) $$ as follows:

  1. 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 $$ \xymatrix{ 0 \ar[r] & A \ar[r] \ar[d] & E' \ar[r] \ar[d] & B' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A \ar[r] & E \ar[r] & B \ar[r] & 0 } $$ The extension $E'$ is called the pullback of $E$ via $B' \to B$.
  2. 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 of short exact sequences $$ \xymatrix{ 0 \ar[r] & A \ar[r] \ar[d] & E \ar[r] \ar[d] & B \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A' \ar[r] & E' \ar[r] & B \ar[r] & 0 } $$ The extension $E'$ is called the pushout of $E$ via $A \to A'$.

To see that this defines a functor as indicated above there are several things to verify. First of all functoriality in the variable $B$ requires that $(E \times_B B') \times_{B'} B'' = E \times_B B''$ which is a general property of fibre products. Dually one deals with functoriality in the variable $A$. Finally, given $A \to A'$ and $B' \to B$ we have to show that $$ A' \amalg_A (E \times_B B') \cong (A' \amalg_A E)\times_B B' $$ as extensions of $B'$ by $A'$. Recall that $A' \amalg_A E$ is a quotient of $A' \oplus E$. Thus the right hand side is a quotient of $A' \oplus E \times_B B'$, and it is straightforward to see that the kernel is exactly what you need in order to get the left hand side.

Note that if $E_1$ and $E_2$ are extensions of $B$ by $A$, then $E_1\oplus E_2$ is an extension of $B \oplus B$ by $A\oplus A$. We pull back by the diagonal map $B \to B \oplus B$ and we push out by the sum map $A \oplus A \to A$ to get an extension $E_1 + E_2$ of $B$ by $A$. $$ \xymatrix{ 0 \ar[r] & A \oplus A \ar[r] \ar[d]^{\sum} & E_1 \oplus E_2 \ar[r] \ar[d] & B \oplus B \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A \ar[r] & E' \ar[r] & B \oplus B \ar[r] & 0\\ 0 \ar[r] & A \ar[r] \ar[u] & E_1 + E_2 \ar[r] \ar[u] & B \ar[r] \ar[u]^{\Delta} & 0 } $$ The extension $E_1 + E_2$ is called the Baer sum of the given extensions.

Lemma 12.6.3. The construction $(E_1, E_2) \mapsto E_1 + E_2$ above defines a commutative group law on $\mathop{\rm Ext}\nolimits_\mathcal{A}(B, A)$ which is functorial in both variables.

Proof. Omitted. $\square$

Lemma 12.6.4. Let $\mathcal{A}$ be an abelian category. Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence in $\mathcal{A}$.

  1. There is a canonical six term exact sequence of abelian groups $$ \xymatrix{ 0 \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(M_3, N) \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(M_2, N) \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(M_1, N) \ar[lld] \\ & \mathop{\rm Ext}\nolimits_\mathcal{A}(M_3, N) \ar[r] & \mathop{\rm Ext}\nolimits_\mathcal{A}(M_2, N) \ar[r] & \mathop{\rm Ext}\nolimits_\mathcal{A}(M_1, N) } $$ for all objects $N$ of $\mathcal{A}$, and
  2. there is a canonical six term exact sequence of abelian groups $$ \xymatrix{ 0 \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(N, M_1) \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(N, M_2) \ar[r] & \mathop{\rm Hom}\nolimits_\mathcal{A}(N, M_3) \ar[lld] \\ & \mathop{\rm Ext}\nolimits_\mathcal{A}(N, M_1) \ar[r] & \mathop{\rm Ext}\nolimits_\mathcal{A}(N, M_2) \ar[r] & \mathop{\rm Ext}\nolimits_\mathcal{A}(N, M_3) } $$ for all objects $N$ of $\mathcal{A}$.

Proof. Omitted. Hint: The boundary maps are defined using either the pushout or pullback of the given short exact sequence. $\square$

    The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 1030–1211 (see updates for more information).

    \section{Extensions}
    \label{section-extensions}
    
    \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
    $$
    0 \to A \to E \to B \to 0.
    $$
    \end{definition}
    
    \noindent
    By abuse of language we often omit mention of the
    morphisms $A \to E$ and $E \to B$, although they are
    definitively part of the structure of an extension.
    
    \begin{definition}
    \label{definition-ext-group}
    Let $\mathcal{A}$ be an abelian category.
    Let $A, B \in \Ob(\mathcal{A})$.
    The set of isomorphism classes of extensions
    of $B$ by $A$ is denoted
    $$
    \Ext_\mathcal{A}(B, A).
    $$
    This is called the {\it $\Ext$-group}.
    \end{definition}
    
    \noindent
    This definition works, because by our conventions
    $\mathcal{A}$ is a set, and hence
    $\Ext_\mathcal{A}(B, A)$ is a set.
    In any of the cases of ``big'' abelian categories
    listed in Categories, Remark \ref{categories-remark-big-categories}.
    one can check by hand that $\Ext_\mathcal{A}(B, A)$
    is a set as well. Also, we will see later that this is
    always the case when $\mathcal{A}$ has either enough projectives
    or enough injectives. Insert future reference here.
    
    \medskip\noindent
    Actually we can turn $\Ext_\mathcal{A}(-, -)$ into a
    functor
    $$
    \mathcal{A} \times \mathcal{A}^{opp} \longrightarrow \textit{Sets}, \quad
    (A, B) \longmapsto \Ext_\mathcal{A}(B, A)
    $$
    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
    $$
    \xymatrix{
    0 \ar[r] & A \ar[r] \ar[d] & E' \ar[r] \ar[d] & B' \ar[r] \ar[d] & 0 \\
    0 \ar[r] & A \ar[r] & E \ar[r] & B \ar[r] & 0
    }
    $$
    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
    of short exact sequences
    $$
    \xymatrix{
    0 \ar[r] & A \ar[r] \ar[d] & E \ar[r] \ar[d] & B \ar[r] \ar[d] & 0 \\
    0 \ar[r] & A' \ar[r] & E' \ar[r] & B \ar[r] & 0
    }
    $$
    The extension $E'$ is called the {\it pushout of $E$ via
    $A \to A'$}.
    \end{enumerate}
    To see that this defines a functor as indicated above
    there are several things to verify. First of all
    functoriality in the variable $B$ requires that
    $(E \times_B B') \times_{B'} B'' = E \times_B B''$
    which is a general property of fibre products.
    Dually one deals with functoriality in the
    variable $A$. Finally, given $A \to A'$ and
    $B' \to B$ we have to show that
    $$
    A' \amalg_A (E \times_B B')
    \cong
    (A' \amalg_A E)\times_B B'
    $$
    as extensions of $B'$ by $A'$. Recall that $A' \amalg_A E$
    is a quotient of $A' \oplus E$.
    Thus the right hand side is a quotient of
    $A' \oplus E \times_B B'$, and it is straightforward to see that
    the kernel is exactly what you need in order to
    get the left hand side.
    
    \medskip\noindent
    Note that if $E_1$ and $E_2$ are extensions of
    $B$ by $A$, then $E_1\oplus E_2$ is an extension
    of $B \oplus B$ by $A\oplus A$. We pull back by
    the diagonal map $B \to B \oplus B$ and we push
    out by the sum map $A \oplus A \to A$ to get
    an extension $E_1 + E_2$ of $B$ by $A$.
    $$
    \xymatrix{
    0 \ar[r] &
    A \oplus A \ar[r] \ar[d]^{\sum} &
    E_1 \oplus E_2 \ar[r] \ar[d] &
    B \oplus B \ar[r] \ar[d] &
    0 \\
    0 \ar[r] &
    A \ar[r] &
    E' \ar[r] &
    B \oplus B \ar[r] &
    0\\
    0 \ar[r] &
    A \ar[r] \ar[u] &
    E_1 + E_2 \ar[r] \ar[u] &
    B \ar[r] \ar[u]^{\Delta} &
    0
    }
    $$
    The extension $E_1 + E_2$ is called the {\it Baer sum} of the
    given extensions.
    
    \begin{lemma}
    \label{lemma-baer-sum}
    The construction $(E_1, E_2) \mapsto E_1 + E_2$
    above defines a commutative group
    law on $\Ext_\mathcal{A}(B, A)$ which is
    functorial in both variables.
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-six-term-sequence-ext}
    Let $\mathcal{A}$ be an abelian category.
    Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence
    in $\mathcal{A}$.
    \begin{enumerate}
    \item There is a canonical six term exact sequence of abelian groups
    $$
    \xymatrix{
    0 \ar[r] &
    \Hom_\mathcal{A}(M_3, N) \ar[r] &
    \Hom_\mathcal{A}(M_2, N) \ar[r] &
    \Hom_\mathcal{A}(M_1, N) \ar[lld] \\
    & \Ext_\mathcal{A}(M_3, N) \ar[r] &
    \Ext_\mathcal{A}(M_2, N) \ar[r] &
    \Ext_\mathcal{A}(M_1, N)
    }
    $$
    for all objects $N$ of $\mathcal{A}$, and
    \item there is a canonical six term exact sequence of abelian groups
    $$
    \xymatrix{
    0 \ar[r] &
    \Hom_\mathcal{A}(N, M_1) \ar[r] &
    \Hom_\mathcal{A}(N, M_2) \ar[r] &
    \Hom_\mathcal{A}(N, M_3) \ar[lld] \\
    & \Ext_\mathcal{A}(N, M_1) \ar[r] &
    \Ext_\mathcal{A}(N, M_2) \ar[r] &
    \Ext_\mathcal{A}(N, M_3)
    }
    $$
    for all objects $N$ of $\mathcal{A}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted. Hint: The boundary maps are defined using either the pushout
    or pullback of the given short exact sequence.
    \end{proof}

    Comments (4)

    Comment #372 by Fan on December 1, 2013 a 7:28 pm UTC

    It might be better to point out that in proof of functorality of Ext, the exactness of the new sequence produced follows from Lemma 3.24.

    Comment #383 by Johan (site) on December 4, 2013 a 4:24 pm UTC

    @#372: Well, I think here the discussion assumes the reader has read the basic facts about short exact sequences (including the snake lemma, etc). If you want to make it more precise, then please code up a lemma with complete proof (but I think we should leave a perhaps shortened informal discussion there).

    Comment #2437 by Raymond Cheng on February 20, 2017 a 9:42 pm UTC

    When turning $\mathrm{Ext}_\mathcal{A}(-,-)$ into a bifunctor, you may wish to swap the $A$ and $B$: write $(B,A) \mapsto \mathrm{Ext}_\mathcal{A}(B,A)$ rather than $(A,B) \mapsto \mathrm{Ext}_\mathcal{A}(A,B)$ to be consistent with the result of the page.

    Comment #2480 by Johan (site) on April 13, 2017 a 10:23 pm UTC

    Thanks, fixed here.

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