Corollary 1

Let A be a ring, J an ideal of A, C a complex of A-modules bounded on the left and p an integer. Suppose that one has JH_m(C) = 0 and prof_A(J ; C_m) > m - p for m \geq p. Then one has H_n(C) = 0 for n \geq p.

In fact for n \geq p the annihilator J_n of H_n(C) contains J, hence one has prof_A(J_n ; C_m) \geq prof_A(J ; C_m) (No. 1, cor. 1 of prop. 2), so that the hypothesis of the proposition is satisfied.