Ext A LA YONEDA WITHOUT THE SCHANUEL LEMMA

By means of a more precise treatment of the congruence relation between n-extensions we show that it is possible to avoid the Schanuel Lemma for proving some results on the functors Ext. An n-extension E from D to C is an exact sequence of the form (1) 0 C El E2 * * En D 0 in an abelian category. The idea to use congruence classes n-extensions as elements of Ext' (D C) is due to Yoneda ([7], [8]; cf. also Buchsbaum [4] and Mac Lane [5, Chapter III]).' In studying Ext' from this point of view, the Schanuel Lemma (Mitchell [6, Chapter VII, Lemma 4.1]) plays an essential role: firstly, in order to get the exactness of the long sequence (Mitchell 1.c. ?5), secondly, for proving that a congruence between two n-extensions can be realized in two steps (Brinkmann [2]). However the Schanuel Lemma and its proof are of a highly technical nature. Therefore, in the following we propose a more straightforward way to these results, the key to which lies in the following THEOREM. Two n-extensions El and E2 are congruent over an n-extension if and only if they are congruent under an n-extension. Here "congruence of El and E2 over (under) an n-extension" means the existence of an n-extension E and of morphisms f' (i = 1, 2) with fixed ends (in the sense of Mitchell 1.c. ?3) forming the diagram