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

\cdots\rightarrow A_{n+1}\buildrel{f_{n+1}}\over{\longrightarrow}A_{n}% \buildrel{f_{n}}\over{\longrightarrow}A_{n-1}\rightarrow\cdots

is said to be exact if each pair of adjacent homomorphisms $(f_{n+1},f_{n})$ is exact – in other words if ${\rm im}f_{n+1}={\rm ker}f_{n}$ 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 $\mathcal{A}$ , where $A_{i}$ and $f_{i}$ above are objects and morphisms in $\mathcal{A}$ .

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