I thought this would be a simple application of the snake lemma, but I cannot do it all. Does anyone know if it is true and, if not, why it is true for tensor product? Suppose X is an abelian category and T:X --> Ab is a right exact functor. Say an object E is T-effaceable if for every exact sequence 0 --> A --> B --> E --> 0, the induced TA --> TB is injective. Now suppose that 0 --> E' --> E --> E'' --> 0 is an exact sequence in which E and E'' are T-effaceable. Does it follow that E' is? Does it help if you suppose that every object has an effaceable cover (and therefore an effaceable resolution)? Then you could define homology, but you want it independent of the resolution and that is where this question actually comes from, trying to prove independence of resolution. Michael