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In homological algebra, an exact functor is a functor that preserves exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that ''fail'' to be exact, but in ways that can still be controlled. == Definitions == Let ''P'' and ''Q'' be abelian categories, and let be a covariant additive functor (so that, in particular, ''F(0)=0''). Let :''0''→''A''→''B''→''C''→''0'' be a short exact sequence of objects in ''P''. We say that ''F'' is * half-exact if ''F(A)''→''F(B)''→''F(C)'' is exact. This is similar to the notion of a topological half-exact functor. * left-exact if ''0''→''F(A)''→''F(B)''→''F(C)'' is exact. * right-exact if ''F(A)''→''F(B)''→''F(C)''→''0'' is exact. * exact if ''0''→''F(A)''→''F(B)''→''F(C)''→''0'' is exact. If ''G'' is a contravariant additive functor from ''P'' to ''Q'', we can make a similar set of definitions. We say that ''G'' is * half-exact if ''G(C)''→''G(B)''→''G(A)'' is exact. * left-exact if ''0''→''G(C)''→''G(B)''→''G(A)'' is exact. * right-exact if ''G(C)''→''G(B)''→''G(A)''→''0'' is exact. * exact if ''0''→''G(C)''→''G(B)''→''G(A)''→''0'' is exact. It is not always necessary to start with an entire short exact sequence ''0''→''A''→''B''→''C''→''0'' to have some exactness preserved; it is only necessary that part of the sequence is exact. The following statements are equivalent to the definitions above: * ''F'' is left-exact if ''0''→''A''→''B''→''C'' exact implies ''0''→''F(A)''→''F(B)''→''F(C)'' exact. * ''F'' is right-exact if ''A''→''B''→''C''→''0'' exact implies ''F(A)''→''F(B)''→''F(C)''→''0'' exact. * ''G'' is left-exact if ''A''→''B''→''C''→''0'' exact implies ''0''→''G(C)''→''G(B)''→''G(A)'' exact. * ''G'' is right-exact if ''0''→''A''→''B''→''C'' exact implies ''G(C)''→''G(B)''→''G(A)''→''0'' exact. Note, that this does not work for half-exactness. The corresponding condition already implies exactness, since you can apply it to exact sequences of the form ''0''→''A''→''B''→''C'' and ''A''→''B''→''C''→0. Thus we get: * ''F'' is exact if and only if ''A''→''B''→''C'' exact implies ''F(A)''→''F(B)''→''F(C)'' exact. * ''G'' is exact if and only if ''A''→''B''→''C'' exact implies ''G(C)''→''G(B)''→''G(A)'' exact. 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「Exact functor」の詳細全文を読む スポンサード リンク
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