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\begin{document}

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\section*{split exact sequence}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{InAbelianCategory}{In an abelian category}\dotfill \pageref*{InAbelianCategory} \linebreak
\noindent\hyperlink{in_a_semiabelian_category}{In a semi-abelian category}\dotfill \pageref*{in_a_semiabelian_category} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToChainHomotopy}{Relation to chain homotopy}\dotfill \pageref*{RelationToChainHomotopy} \linebreak
\noindent\hyperlink{OfVectorSpaces}{Of free modules and vector spaces}\dotfill \pageref*{OfVectorSpaces} \linebreak
\noindent\hyperlink{InvolvingInjectiveObjects}{Involving injective/projective objects}\dotfill \pageref*{InvolvingInjectiveObjects} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{InAbelianCategory}{}\subsubsection*{{In an abelian category}}\label{InAbelianCategory}

Let $\mathcal{A}$ be an [[abelian category]].

\begin{defn}
\label{SplitnessInAbelianCategory}\hypertarget{SplitnessInAbelianCategory}{}
A [[short exact sequence]] $0\to A \stackrel{i}{\to} B \stackrel{p}{\to} C\to 0$ in $\mathcal{A}$ is called \textbf{split} if either of the following equivalent conditions hold

\begin{enumerate}%
\item There exists a [[section]] of $p$, hence a morphism $s \colon C\to B$ such that $p \circ s = id_C$.


\item There exists a [[retract]] of $i$, hence a morphism $r \colon B\to A$ such that $r \circ i = id_A$.


\item There exists an [[isomorphism]] of sequences with the sequence

\begin{displaymath}
0\to A\to A\oplus C\to C\to 0
\end{displaymath}
given by the [[direct sum]] and its canonical injection/projection morphisms.



\end{enumerate}
\end{defn}
\begin{lemma}
\label{SplittingLemma}\hypertarget{SplittingLemma}{}
\textbf{(splitting lemma)}

The three conditions in def. \ref{SplitnessInAbelianCategory} are indeed [[equivalence|equivalent]].

\end{lemma}
\begin{proof}
It is clear that the third condition implies the first two: take the section/retract to be given by the canonical injection/projection maps that come with a [[direct sum]].

Conversely, suppose we have a retract $r \colon B \to A$ of $i \colon A \to B$. Write $P \colon B \stackrel{r}{\to} A \stackrel{i}{\to} B$ for the corresponding [[idempotent]].

Then every element $b \in B$ can be decomposed as $b = (b - P(b)) + P(b)$ hence with $b - P(b) \in ker(r)$ and $P(b) \in im(i)$. Moreover this decomposition is unique since if $b = i(a)$ while at the same time $r(b) = 0$ then $0 = r(i(a)) = a$. This shows that $B \simeq im(i) \oplus ker(r)$ is a [[direct sum]] and that $i \colon A \to B$ is the canonical inclusion of $im(i)$. By exactness it then follows that $ker(r) \simeq im(p)$ and hence that $B \simeq A \oplus C$ with the canonical inclusion and projection.

The implication that the second condition also implies the third is formally dual to this argument.

\end{proof}
\hypertarget{in_a_semiabelian_category}{}\subsubsection*{{In a semi-abelian category}}\label{in_a_semiabelian_category}

There is a nonabelian analog of split exact sequences in [[semiabelian categories]]. See there.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToChainHomotopy}{}\subsubsection*{{Relation to chain homotopy}}\label{RelationToChainHomotopy}

\begin{prop}
\label{}\hypertarget{}{}
A [[long exact sequence]] $C_\bullet$ is \emph{split exact} precisely if the [[weak homotopy equivalence]] from the 0-chain complex, namely the [[quasi-isomorphism]] $0 \to C_\bullet$ is actually a [[chain homotopy]] [[homotopy equivalence|equivalence]], in that the [[identity]] on $C_\bullet$ has a [[null homotopy]].

\end{prop}
\hypertarget{OfVectorSpaces}{}\subsubsection*{{Of free modules and vector spaces}}\label{OfVectorSpaces}

\begin{prop}
\label{}\hypertarget{}{}
Every exact sequence of [[free abelian groups]] is split, assuming the [[axiom of choice]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Every exact sequence of [[free modules]] which is bounded below is split.

\end{prop}
Let $k$ be a [[field]] and denote by $\mathcal{A} \coloneqq k$[[Vect]] the [[category]] of [[vector spaces]] over $k$.

\begin{cor}
\label{}\hypertarget{}{}
Every [[short exact sequence]] of vector spaces is split.

\end{cor}
\hypertarget{InvolvingInjectiveObjects}{}\subsubsection*{{Involving injective/projective objects}}\label{InvolvingInjectiveObjects}

\begin{lemma}
\label{}\hypertarget{}{}
If in a [[short exact sequence]] $0 \to A \to B \to C \to 0$ in an [[abelian category]] the first object $A$ is an [[injective object]] or the last object is a [[projective object]] then the sequence is split exact.

\end{lemma}
\begin{proof}
Consider the first case. The other is formally dual.

By the properties of a [[short exact sequence]] the morphism $A \to B$ here is a [[monomorphism]]. By definition of [[injective object]], if $A$ is injective then it has the [[right lifting property]] against [[monomorphisms]] and so there is a morphism $q : B \to A$ that makes the following [[diagram]] [[commuting diagram|commute]]:

\begin{displaymath}
\itexarray{
     A &\stackrel{id_A}{\to}& A
     \\
     \downarrow & \nearrow_{q}
     \\
     B
  }
  \,.
\end{displaymath}
Hence $q$ is a [[retract]] as in def. \ref{SplitnessInAbelianCategory}.

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For instance section 1.4 of

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}

\end{itemize}
[[!redirects split sequence]]

[[!redirects split exact sequences]] [[!redirects split sequences]]

[[!redirects split short exact sequence]] [[!redirects split short exact sequences]]

[[!redirects splitting lemma]]



\end{document}