exact sequence

If we have two homomorphisms $f:A\to B$ and $g:B\to C$in some category of modules,then we say that $f$ and $g$ are exact at $B$if the image of $f$ is equal to the kernel of $g$.

A sequence of homomorphisms

$$\mathrm{\cdots }\to A_{n+1}\stackrel{f_{n+1}}{⟶}{A}_n\stackrel{f_n}{⟶}{A}_{n-1}\to \mathrm{\cdots }$$

is said to be exact if each pairof adjacent homomorphisms $(f_{n+1},f_n)$ is exact –in other words if $\mathrm{im}{f}_{n+1}=\ker f_n$ for all $n$.

Compare this to the notion of a chain complex.

Remark. The notion of exact sequences can be generalized to any abelian category $𝒜$, where $A_i$ and $f_i$ above are objects and morphisms in $𝒜$.

Title	exact sequence
Canonical name	ExactSequence
Date of creation	2013-03-22 12:09:27
Last modified on	2013-03-22 12:09:27
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	7
Author	antizeus (11)
Entry type	Definition
Classification	msc 16-00
Related topic	ExactSequence2
Related topic	CategoricalSequence
Related topic	HomologicalComplexOfTopologicalVectorSpaces
Related topic	CategoricalDiagramsAsFunctors
Related topic	SpinGroup
Related topic	AlternativeDefinitionOfAnAbelianCategory