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A lemma in homological algebra

Published online by Cambridge University Press:  24 October 2008

G. M. Kelly
G. M. Kelly
Affiliation:
Tulane University, Louisiana
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 49 - 52
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Cartan, H. and Eilenberg, S.Homological algebra (Princeton, 1956).Google Scholar
(2)Dold, A.Zur Homotopietheorie der Kettenkomplexe. Math. Ann. 140 (1960), 278–298.CrossRefGoogle Scholar
(3)Godement, R.Théorie des faisceaux (Paris, 1958).Google Scholar
(4)MacLane, S.Homology (New York, 1963).CrossRefGoogle Scholar