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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 <math>\mathcal{A}</math> be an [[Abelian category]] with [[enough projectives]], and let ''A''<sub>∗</sub> be a chain complex with objects in <math>\mathcal{A}</math>. Then a '''Cartan–Eilenberg resolution''' of ''A''<sub>∗</sub> is an upper half-plane [[double complex]] ''P''<sub>∗∗</sub> (i.e., ''P''<sub>pq</sub> = 0 for ''q'' < 0) consisting of projective objects of <math>\mathcal{A}</math> and a chain map ε : ''P''<sub>p0</sub> → ''A''<sub>p</sub> such that

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			* Taking the boundaries and homology of the ''p''th column of ''P''<sub>∗∗</sub> give projective resolutions of the ''p''th boundary and homology groups of ''A''<sub>∗</sub>, respectively. In other words, one has chain complexes ε : ''P''<sub>p∗</sub> → ''A''<sub>p</sub>, and applying the boundary and homology functors to these complexes turns them into projective resolutions.

			There is an analogous definition using injective resolutions and cochain complexes.

			The existence of Cartan–Eilenberg resolutions can be proved via the [[horseshoe lemma]].

			=== Hyper-derived functors ===

			Given a right [[exact functor]] <math>F \colon \mathcal{A} \to \mathcal{B}</math>, one can define the left hyper-derived functors of ''F'' on a chain complex ''A''<sub>∗</sub> by constructing a Cartan–Eilenberg resolution ε : ''P''<sub>∗∗</sub> → ''A''<sub>∗</sub>, applying ''F'' to ''P''<sub>∗∗</sub>, and taking the homology of the resulting total complex.

			Similarly, one can also define right hyper-derived functors for left exact functors.

			== See also ==
			* [[Hyperhomology]]

			== References ==
			* {{Citation
			\| last1=Weibel
			\| first1=Charles A.
			\| title=An introduction to homological algebra
			\| publisher=[[Cambridge University Press]]
			\| isbn=978-0-521-55987-4
			\| mr=1269324
			\| year=1994
			\| series=Cambridge Studies in Advanced Mathematics
			\| volume=38
			}}

			{{DEFAULTSORT:Cartan-Eilenberg resolution}}
			[[Category:Homological algebra]]

en>Battymathmo at 08:24, 6 August 2012

2012-08-06T08:24:55Z

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en>Battymathmo at 08:24, 6 August 2012