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In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a complex. It can be used to construct hyper-derived functors. == Definition == Let be an Abelian category with enough projectives, and let ''A''∗ be a chain complex with objects in . Then a Cartan–Eilenberg resolution of ''A''∗ is an upper half-plane double complex ''P''∗∗ (i.e., ''P''pq = 0 for ''q'' < 0) consisting of projective objects of and a chain map ε : ''P''p0 → ''A''p such that * ''A''p = 0 implies that the ''p''th column is zero (''P''pq = 0 for all ''q''). * For any fixed column, * * the kernels of each of the horizontal maps ''starting'' at that column (which themselves form a complex) are in fact exact, * * the same is true for the images of those maps, and * * the same is true for the homology of those maps. (In fact, it would suffice to require it for the kernels and homology - the case of images follows from these.) In particular, since the kernels, cokernels, and homology will all be projective, they will give a projective resolution of the kernels, cokernels, and homology of the original complex ''A•'' There is an analogous definition using injective resolutions and cochain complexes. The existence of Cartan–Eilenberg resolutions can be proved via the horseshoe lemma. 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「Cartan–Eilenberg resolution」の詳細全文を読む スポンサード リンク
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