Let f: A -B be a surjective homomorphism of noetherian local commutative rings that induces an isomorphism between the first Koszul homology modules and an epimorphism between the second Koszul homology modules. Then f induces isomorphisms between Koszul homology modules in all dimensions. Let (A, m, k) be a local noetherian (commutative with unit) ring, I an ideal of A, B = A/I, and n the maximal ideal of B. Let .xl,. . ,x4} be a minimal set of generators of the ideal m of A, and Yi,... , yn the images of these elements in B. Let {v1, ... ., vr} be a minimal set of generators of the ideal n of B. Fix for each 1 < j < n elements bj, E B such that yj = EZ=, bjv. Let E = A(Xi .. . ,Xn;dXi = xi) be the Koszul complex associated to the elements Xl,... , xn of A, and E'B(V1, . . . , V,.; dVi = vi) be the Koszul complex associated to the elements v1, .. ., v, of B. By a little abuse of language we denote H*(A) = H*(E), H*(B) = H*(E') (they do not depend, up to isomorphism, of the minimal set of generators of the maximal ideal). Let f: E -* E' be the homomorphism of complexes extending the projection map A -* B by sending Xi to EZ=1 bjaVa. L. L. Avramov and E. S. Golod [4, Proposition 1] show that if the ideal I is generated by a regular sequence which is part of a minimal system of generators of the ideal m, then H* (f): H* (A) -* H* (B) is an isomorphism. In [8] (see [9, (2.3.6)] S. S. Strogalov shows that the converse also holds. The following result shows that it is enough to consider the first two homology modules: Proposition 1. If H1 (f ) is an isomorphism and H2 (f) is surjective, then the ideal I is generated by a regular sequence which is part of a minimal system of generators of the ideal m. Proof. Let 0 -* U -* F -P m -* 0 be an exact sequence of A-modules with F free with basis{z1, .,Zn} and p(zi) = xi, 1 < i < n, and let 0 -* U' -* F' P n -* 0be an exact sequence of B-modules with F' free with basis {zj, . .. , z' } and p'(z') = vi, 1 < i < r. Let g: F A F -* F, g(a A b) = p(a)b-p(b)a, and V = Im(g). Define similarly g': F'A F' -* F' and V'= Im(g'). We have isomorphisms U/V = H1 (A), U'lV' = H1(B). Received by the editors February 10, 1997 and, in revised form, May 5, 1998. 1991 Mathematics Subject Classification. Primary 13H10, 13D03.