Exercise 14, § 3

Suppose that the ring A is local noetherian; denote by m its maximal ideal. Let L be a complex of finitely generated A-modules, zero in degrees < 0, M a finitely generated A-module, and $q : L \to M$ a morphism. We make the following hypotheses: (i) The homomorphism $1 \otimes q : (A/m) \otimes_A L_0 \to (A/m) \otimes_A M$ is injective. (ii) We have $d_i(L_i) \subset mL_{i-1}$ for $i \geqslant 1$, and the mapping

is injective. Let (P, p) be a projective resolution of M, $u : L \to P$ a morphism such that $p \circ u = q$. Prove that u is injective and that u(L) is a direct factor of P as a graded A-module (reduce to the case where the resolution P is minimal, and use VIII, § 8, No. 3, Cor. 3).