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In mathematics, the cokernel of a linear mapping of vector spaces ''f'' : ''X'' → ''Y'' is the quotient space ''Y''/im(''f'') of the codomain of ''f'' by the image of ''f''. The dimension of the cokernel is called the ''corank'' of ''f''. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain). Intuitively, given an equation ''f(x) = y'' that one is seeking to solve, the cokernel measures the ''constraints'' that ''y'' must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the ''degrees of freedom'' in a solution, if one exists. This is elaborated in intuition, below. More generally, the cokernel of a morphism ''f'' : ''X'' → ''Y'' in some category (e.g. a homomorphism between groups or a bounded linear operator between Hilbert spaces) is an object ''Q'' and a morphism ''q'' : ''Y'' → ''Q'' such that the composition ''q f'' is the zero morphism of the category, and furthermore ''q'' is universal with respect to this property. Often the map ''q'' is understood, and ''Q'' itself is called the cokernel of ''f''. In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism ''f'' : ''X'' → ''Y'' is the quotient of ''Y'' by the image of ''f''. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient. == Formal definition == One can define the cokernel in the general framework of category theory. In order for the definition to make sense the category in question must have zero morphisms. The cokernel of a morphism ''f'' : ''X'' → ''Y'' is defined as the coequalizer of ''f'' and the zero morphism 0''XY'' : ''X'' → ''Y''. Explicitly, this means the following. The cokernel of ''f'' : ''X'' → ''Y'' is an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that the diagram commutes. Moreover the morphism ''q'' must be universal for this diagram, i.e. any other such ''q''′: ''Y'' → ''Q''′ can be obtained by composing ''q'' with a unique morphism ''u'' : ''Q'' → ''Q''′: As with all universal constructions the cokernel, if it exists, is unique up to a unique isomorphism, or more precisely: if ''q'' : ''Y'' → ''Q'' and ''q‘'' : ''Y'' → ''Q‘'' are two cokernels of ''f'' : ''X'' → ''Y'', then there exists a unique isomorphism ''u'' : ''Q'' → ''Q‘'' with ''q‘'' = ''u'' ''q''. Like all coequalizers, the cokernel ''q'' : ''Y'' → ''Q'' is necessarily an epimorphism. Conversely an epimorphism is called ''normal'' (or ''conormal'') if it is the cokernel of some morphism. A category is called ''conormal'' if every epimorphism is normal (e.g. the category of groups is conormal). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cokernel」の詳細全文を読む スポンサード リンク
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