Acyclic Models and Triples

  • Michael Barr
  • Jon Beck


We shall prove two theorems on the “triple” cohomology of algebras [1] using a method of acyclic models suggested by H. Appelgate. Specifically, we show that the triple cohomology coincides with slight modifications of the usual theories (the same modifications as used in [8] in the cases of groups and associative algebras). We also prove a direct sum theorem for the cohomology of a coproduct of algebras, subject to a certain condition.


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  1. [1]
    Beck, J.: Triples, algebras and cohomology. To appear.Google Scholar
  2. [2]
    Eilenberg, S., and S. MacLane: Acyclic models. Amer. J. Math. 75, 189–199 (1953).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    — Cohomology theory in abstract groups, I. Ann. Math. 48, 51–78 (1947).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    —, and J. C. Moore: Adjoint functors and triples. III. J. Math. 9, 381–398 (1965).MathSciNetMATHGoogle Scholar
  5. [5]
    Harrison, D. K.: Commutative algebras and cohomology. Trans. AMS, 104, 191–204 (1962).MATHCrossRefGoogle Scholar
  6. [6]
    Lyndon, R. C.: New proof for a theorem of Eilenberg and MacLane. Ann. Math. 50, 731–735 (1949).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    MacLane, S.: Homology. Berlin: Springer 1963.MATHGoogle Scholar
  8. [8]
    Barr, M., and G. Rinehart: Cohomology as the derived functor of derivations. To appear in Trans. AIMS.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1966

Authors and Affiliations

  • Michael Barr
    • 1
    • 2
  • Jon Beck
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

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