# 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