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'exact sequence'
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| Title of object: |
exact sequence |
| Canonical Name: |
ExactSequence2 |
| Type: |
Definition |
| Created on: |
2002-04-24 01:12:40 |
| Modified on: |
2004-03-14 16:17:27 |
| Classification: |
msc:18E10 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb,amsthm}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
\usepackage[all]{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\A}{\mathcal{A}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\cok}{\operatorname{cok}} |
Content:
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
Let $\A$ be an abelian category. We begin with a preliminary definition.
\begin{definition}
For any morphism $f: A \longrightarrow B$ in $\A$, let $m: X \longrightarrow B$ be the morphism equal to $\ker(\cok(f))$. Then the object $X$ is called the {\em image} of $f$, and denoted $\Im(f)$. The morphism $m$ is called the {\em image morphism} of $f$, and denoted $\im(f)$.
\end{definition}
Note that $\Im(f)$ is not the same as $\im(f)$: the former is an object of $\A$, while the latter is a morphism of $\A$. We note that $f$ factors through $\im(f)$:
$$
\xymatrix{
A \ar[r]^-{e} \ar@/_1pc/[rr]_{f} & \Im(f) \ar[r]^-{\im(f)} & B
}
$$
The proof is as follows: by definition of cokernel, $\cok(f) f = 0$; therefore by definition of kernel, the morphism $f$ factors through $\ker(\cok(f)) = \im(f)$, and this factor is the morphism $e$ above. Furthermore $m$ is a monomorphism and $e$ is an epimorphism, although we do not prove these facts.
\begin{definition}
A sequence
$$
\xymatrix{
\cdots A \ar[r]^-f & B \ar[r]^-g & C \cdots
}
$$
of morphisms in $\A$ is {\em exact} at $B$ if $\ker(g) = \im(f)$.
\end{definition} |
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