Journal of Pure and Applied Algebra 6 (1975) 133-- 153 0 North-Holland Publishing Company zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA THE BIVARIANT LONG EXACT SEQUENCE FOR THE EXT FUNCTOR I.S. PRESSMAN Carkton UGversity, Ottawa, Ont., Cartada Communicated by Peter liilton Reccivcd September 11, 1974 0. Introduction It is traditional to present the Ext functor in two distinct formats: as a covariant functor on the second variable with the first held fixed, and as a contravariant functor on the first variable with the second held fixed. In this paper, the situation where both variables are allowed to vary simultaneously is described, so that Ext acts in a bivariarzt manner. If one has two short exact sequences P 4,:0-+/l-%B--+C-+O, ~:O+X-LY -L&+0 one proves (Theorem 2.7) that there is a long exact sequence s(Q, ct) : 0 -+ Hom(Z,A) X2 HomU’&) a Hom(%& 70, &‘(Z,A) 11, Ext*(Y,B) -?‘+ Ext’(y,@ -‘* -.a which generalizes the usual two long exact sequences in that for proper choices of Q) or Q one obtains either the contravariant or covariant long exact sequences together with a commutative diagram to the bivariant sequence from each of the others (see 3-l). The groups Ext”(y,& are computed in the category of morphisms of an abelian category. S is a functor from the category of pairs of short exact sequences to the category of long exact sequences of abelian groups which is contravarian t in the first and covariant in the second variable. The morphisms of S are completely des- cribed and are not difficult to use. The elements of Ext n+l(Z,A) which are images of #I, the connecting homomor- phisms, are interpreted in 3.5 as obstructions to factoring commutative squares, and in 3.6 as obstructions to factoring diagrams of long exact sequences. 7* can be com- puted with particular ease. Exact squares are studied in Section 4. A five term exact sequence is introduced. which is exact if and only if the square is exact. A square is exact iff it is the middle square of an extension l? of an epimorphism /3by a monomorphism y of length 2