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A lemma in homological algebra

  • G. M. Kelly (a1)
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In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is

(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then

is an isomorphism.

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References
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(1)Cartan, H. and Eilenberg, S.Homological algebra (Princeton, 1956).
(2)Dold, A.Zur Homotopietheorie der Kettenkomplexe. Math. Ann. 140 (1960), 278298.
(3)Godement, R.Théorie des faisceaux (Paris, 1958).
(4)MacLane, S.Homology (New York, 1963).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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