Satellites of half exact functors, a correction

Abstract

In Chapter III of H. Cartan and S. Eilenberg, Homological algebra, there is the substantial THEOREM 3.1. Let (1) O A' A O be an exact sequence. If T is a covariant half exact functor then the sequence (2) * * * Sn-S T(A) SnT(A') SnT(A) Sn T(A) Sn+'T(A') >... is exact. For T contravariant, A' and A should be interchanged. The proof, which occupies pp. 40-42 of the book, is routine until the critical step of showing that the sequence S1T(A)-*T(A') -->T(A) is exact. The proof of this assertion begins with the last paragraph of p. 41 and goes through most of p. 42. At Cambridge University in 1957 (on a National Science Foundation fellowship) we observed that this part of the proof contains an error. Fortunately it can be made right, as we communicated to Professor Eilenberg at the time. Since we have been asked for this correction several times it seems proper to put it in print. The precise mistake is the sentence on lines 12-15 of p. 42. In the special case being considered, the diagram on p. 41 reduces to 0 0