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where G is an explicitly described submodule of Ext^(a, M). (Cf. Harrison [5, Theorem 2].) We then show that for the category of commutative associative algebras over a coefficient field &, the group 8>(A, M) as defined in [4] coincides with that defined by Harrison in [5]. (An example of Barr in a note to appear [ l] shows that in the category of commutative associative algebras, 8(^4, M) is not the first derived functor of the Baer group S(^4, M) when the latter is considered as a functor of the module M.) More generally, two cohomology theories for a category of algebras or groups with sufficiently many projectives coincides if (i) each possesses an exact sequence analogous to (2) with S ^. , M) the derivations of A into M, and (ii) &(A, M)=0 whenever A is projective. In order to be brief, we prove the exactness of (2) explicitly only for commutative associative algebras over fe, but the reader of [4] will observe that the considerations apply to any "category of interest" in the sense of that paper. 1 The author gratefully wishes to acknowledge the support of the Institute for Advanced Study and of the N.S.F. through Grant GP 3683 with the University of Pennsylvania.

The third cohomology group of a ring and the commutative cohomology theory