Published online by Cambridge University Press: 24 October 2008
Hilton and Rees have proved (cf. (1), Theorem 1·3) that every natural map
is induced by a map from A to B (or, Hom (A, B) → Next1,1 (A, B) is surjective). It follows that Ext1 (B, −) and Ext1 (A, −) are naturally isomorphic if and only if A and B are quasi-isomorphic (loc. cit., Theorem 2·6), i.e. if there exist projective objects P, Q and an isomorphism . One can ask whether these theorems remain true for higher extension functors.