exact sequence

If we have two homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f:AB and g:BC in some categoryMathworldPlanetmath 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 sequencePlanetmathPlanetmath of homomorphisms


is said to be exact if each pair of adjacent homomorphisms (fn+1,fn) is exact – in other words if imfn+1=kerfn for all n.

Compare this to the notion of a chain complexMathworldPlanetmath.

Remark. The notion of exact sequencesPlanetmathPlanetmathPlanetmathPlanetmath can be generalized to any abelian categoryMathworldPlanetmathPlanetmathPlanetmath 𝒜, where Ai and fi 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