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\begin{document}

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\section*{exact sequence}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{definition_in_additive_categories}{Definition in additive categories}\dotfill \pageref*{definition_in_additive_categories} \linebreak
\noindent\hyperlink{definition_in_pointed_sets}{Definition in pointed sets}\dotfill \pageref*{definition_in_pointed_sets} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{computing_terms_in_an_exact_sequence}{Computing terms in an exact sequence}\dotfill \pageref*{computing_terms_in_an_exact_sequence} \linebreak
\noindent\hyperlink{ExactnessAndQuasiIsomorphisms}{Exactness and quasi-isomorphisms}\dotfill \pageref*{ExactnessAndQuasiIsomorphisms} \linebreak
\noindent\hyperlink{SESAndQuotients}{Short exact sequences and quotients}\dotfill \pageref*{SESAndQuotients} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SpecificExamples}{Specific examples}\dotfill \pageref*{SpecificExamples} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \textbf{exact sequence} may be defined in a [[semi-abelian category]], and more generally in a [[homological category]]. It is a [[sequential diagram]] in which the [[image]] of each [[morphism]] is equal to the [[kernel]] of the next morphism.

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

\hypertarget{definition_in_additive_categories}{}\subsubsection*{{Definition in additive categories}}\label{definition_in_additive_categories}

Let $\mathcal{A}$ be an [[additive category]] (often assumed to be an [[abelian category]], for instance $\mathcal{A} = R$[[Mod]] for $R$ some [[ring]]).

\begin{defn}
\label{ExactSequence}\hypertarget{ExactSequence}{}
An \textbf{exact sequence} in $\mathcal{A}$ is a [[chain complex]] $C_\bullet$ in $\mathcal{A}$ with vanishing [[chain homology]] in each degree:

\begin{displaymath}
\forall n \in \mathbb{N} . H_n(C) = 0 
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{ShortExactSequence}\hypertarget{ShortExactSequence}{}
A \textbf{short exact sequence} is an exact sequence, def. \ref{ExactSequence} of the form

\begin{displaymath}
\cdots \to 0 \to 0 \to A \to B \to C \to 0 \to 0 \to \cdots
  \,.
\end{displaymath}
One usually writes this just ``$0 \to A \to B \to C \to 0$'' or even just ``$A \to B \to C$''.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A general exact sequence is sometimes called a \textbf{long exact sequence}, to distinguish from the special case of a short exact sequence.

\end{remark}
\begin{prop}
\label{CharacterizationByMonoEpi}\hypertarget{CharacterizationByMonoEpi}{}
Explicitly, a sequence of morphisms

\begin{displaymath}
0 \to A \stackrel{i}\to B \stackrel{p}\to C  \to 0
\end{displaymath}
is short exact, def. \ref{ShortExactSequence}, precisely if

\begin{enumerate}%
\item $i$ is a [[monomorphism]],


\item $p$ is an [[epimorphism]],


\item and the [[image]] of $i$ equals the [[kernel]] of $p$ (equivalently, the [[coimage]] of $p$ equals the [[cokernel]] of $i$).



\end{enumerate}
\end{prop}
\begin{proof}
The third condition is the definition of exactness at $B$. So we need to show that the first two conditions are equivalent to exactness at $A$ and at $C$.

This is easy to see by looking at elements when $\mathcal{A} \simeq R$[[Mod]], for some ring $R$ (and the general case can be reduced to this one using one of the \href{abelian%20category#EmbeddingTheorems}{embedding theorems}):

The sequence being exact at

\begin{displaymath}
0 \to A \to B
\end{displaymath}
means, since the [[image]] of $0 \to A$ is just the element $0 \in A$, that the [[kernel]] of $A \to B$ consists of just this element. But since $A \to B$ is a [[group homomorphism]], this means equivalently that $A \to B$ is an [[injection]].

Dually, the sequence being exact at

\begin{displaymath}
B \to C \to 0
\end{displaymath}
means, since the [[kernel]] of $C \to 0$ is all of $C$, that also the [[image]] of $B \to C$ is all of $C$, hence equivalently that $B \to C$ is a [[surjection]].

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{[[split exact sequence]]} is a short exact sequence as above in which $i$ is a [[split monomorphism]], or (equivalently) in which $p$ is a [[split epimorphism]].

\end{defn}
In this case, $B$ may be decomposed as the [[biproduct]] $A \oplus C$ (with $i$ and $p$ the usual biproduct inclusion and projection); this sense in which $B$ is `split' into $A$ and $C$ is the origin of the general terms `split (mono/epi)morphism'.

\hypertarget{definition_in_pointed_sets}{}\subsubsection*{{Definition in pointed sets}}\label{definition_in_pointed_sets}

It is also helpful to consider a similar notion in the case of a [[pointed set]].

\begin{defn}
\label{}\hypertarget{}{}
In the category $Set_*$ of [[pointed sets]], a sequence

\begin{displaymath}
\itexarray{
  (A, a) & \overset{f}{\to} & (B, b) & \overset{g}{\to} & (C, c)
}
\end{displaymath}
is said to be \textbf{exact} at $(B, b)$ if $im f = g^{-1}(c)$.

For [[concrete category|concrete]] pointed categories (ie. a category $\mathcal{C}$ with a faithful functor $F: \mathcal{C} \to Set_*$), a sequence is exact if the image under $F$ is exact.

\end{defn}
In the case of (abelian) categories like $Ab$ and $R-Mod$, the two notions of exactness coincide if we pick the point of each group/module to be $0$. Such a general notion is useful in cases such as the [[long exact sequence of homotopy groups]] where the homotopy ``groups'' for small $n$ are just pointed sets without a group structure.

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

\hypertarget{computing_terms_in_an_exact_sequence}{}\subsubsection*{{Computing terms in an exact sequence}}\label{computing_terms_in_an_exact_sequence}

A typical use of a long exact sequence, notably of the [[homology long exact sequence]], is that it allows to determine some of its entries in terms of others.

The characterization of short exact sequences in prop. \ref{CharacterizationByMonoEpi} is one example for this: whenever in a long exact sequence one entry vanishes as in $\cdot \to 0 \to C_n \to \cdot$ or $\cdot \to C_n \to 0 \to \cdots$, it follows that the \emph{next} morphism out of or into the vanishing entry is a [[monomorphism]] or [[epimorphism]], respectively.

In particular:

\begin{prop}
\label{}\hypertarget{}{}
If part of an exact sequence looks like

\begin{displaymath}
\cdots \to 0 \to C_{n+1} \stackrel{\partial_n}{\to} C_n \to 0 \to \cdots
  \,,
\end{displaymath}
then $\partial_n$ is an [[isomorphism]] and hence

\begin{displaymath}
C_{n+1} \simeq C_n
  \,.
\end{displaymath}
\end{prop}
\hypertarget{ExactnessAndQuasiIsomorphisms}{}\subsubsection*{{Exactness and quasi-isomorphisms}}\label{ExactnessAndQuasiIsomorphisms}

\begin{prop}
\label{}\hypertarget{}{}
A chain complex $C_\bullet$ is exact (is a long exact sequence), precisely if the unique [[chain map]] from the [[initial object]], the 0-complex

\begin{displaymath}
0 \to C_\bullet
\end{displaymath}
is a [[quasi-isomorphism]].

\end{prop}
\hypertarget{SESAndQuotients}{}\subsubsection*{{Short exact sequences and quotients}}\label{SESAndQuotients}

The following are some basic lemmas that show how given a short exact sequence one obtains new short exact sequences from forming [[quotients]]/[[cokernels]] (see \hyperlink{Wise}{Wise}).

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

\begin{lemma}
\label{}\hypertarget{}{}
For

\begin{displaymath}
A \to B \to C \to 0
\end{displaymath}
an exact sequence in $\mathcal{A}$ and for $X \to B$ any morphism in $\mathcal{A}$, also

\begin{displaymath}
A \to B/X \to C/X \to 0
\end{displaymath}
is a short exact sequence.

\end{lemma}
\begin{proof}
We have an exact sequence of complexes of length 2

\begin{displaymath}
\itexarray{
    0 &\to& X &\stackrel{id}{\to}& X &\to& 0
    \\
    \downarrow & & \downarrow & & \downarrow & & \downarrow
    \\
    A &\to& B &\to& C &\to& 0
  }
\end{displaymath}
and the exact sequence to be demonstrated is degreewise the [[cokernel]] of this sequence. So the statement reduces to the fact that forming cokernels is a [[right exact functor]].

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
For

\begin{displaymath}
0 \to A \to B \to C
\end{displaymath}
an exact sequence and $X \to A$ any morphism, also

\begin{displaymath}
0 \to A/X \to B/X  \to C
\end{displaymath}
is exact.

\end{lemma}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{SpecificExamples}{}\subsubsection*{{Specific examples}}\label{SpecificExamples}

\begin{example}
\label{MultiplicationAndCyclicGroup}\hypertarget{MultiplicationAndCyclicGroup}{}
Let $\mathcal{A} = \mathbb{Z}$[[Mod]] $\simeq$ [[Ab]]. For $n \in \mathbb{N}$ with $n \geq 1$ let $\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}$ be the [[linear map]]/[[homomorphism]] of [[abelian groups]] which acts by the ordinary multiplication of [[integers]] by $n$. This is clearly an [[injection]]. The [[cokernel]] of this morphism is the projection to the [[quotient group]], which is the [[cyclic group]] $\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}$. Hence we have a short exact sequence

\begin{displaymath}
0 \to \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}
  \to \mathbb{Z}_n
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
The [[connecting homomorphism]] of the [[long exact sequence in homology]] induced from short exact sequences of the form in example \ref{MultiplicationAndCyclicGroup} is called a \emph{[[Bockstein homomorphism]]}.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item [[exponential exact sequence]]


\item [[Kummer sequence]]


\item [[Artin-Schreier sequence]]


\item [[Kummer-Artin-Schreier-Witt exact sequence]]



\end{itemize}
\end{example}
\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

\begin{itemize}%
\item [[long exact sequence of homotopy groups]]


\item [[long exact sequence in cohomology]]


\item [[long exact sequence in homology]]


\item [[Serre long exact sequence]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[splicing of short exact sequences]]


\item [[exact functor]]


\item [[exact sequence of Hopf algebras]]



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

A standard introduction is for instance in section 1.1 of

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

\end{itemize}
The quotient lemmas from \hyperlink{SESAndQuotients}{above} are discussed in

\begin{itemize}%
\item [[Jonathan Wise]], \emph{The Snake Lemma} (\href{http://math.stanford.edu/~jonathan/papers/snake.pdf}{pdf})

\end{itemize}
in the context of the [[salamander lemma]] and the [[snake lemma]].

[[!redirects exact sequences]]

[[!redirects long exact sequence]] [[!redirects short exact sequence]]

[[!redirects long exact sequences]] [[!redirects short exact sequences]]



\end{document}