Bivariant long exact sequences. II (Q909015)

[For part I see Quaest. Math. 9, 207-226 (1986; Zbl 0606.18005).] Given a pair of short exact sequences \[ 1)\quad 0\to X\to^{\gamma}Y\to^{\delta}Z\to 0,\quad 0\to A\to^{\alpha}B\to^{\beta}C\to 0 \] in an abelian category A, with sufficiently many projectives and injectives, and given an additive bifunctor T we show that T applied to the pair (1) gives rise to a diagram of a type studied by C. T. C. Wall that contains 15 interlocking long exact sequences involving the derived functors of T at (A,X), (A,Y), etc., and also involving the derived functors of \(T_ P\) and \(T_ Q\) which are two functors with domain the category of arrows of A, \(A^ 2\), that arise through the failure of T to preserve pullbacks and pushouts. In the case of Hom (respectively \(\otimes)\) in the category of G-modules for a group G the derived functors of \(T_ P\) (respectively \(T_ Q)\) are expressed in terms of group cohomology (respectively homology).