Exercise 11, § 9

Let $A$ be a noetherian ring.

a) Suppose that $A$ is local. Let $P, Q$ be bounded flat $A$-complexes. Suppose that $H(P \otimes_A Q)$ is zero in degree $\neq 0$, and free of rank 1 in degree 0. Prove that there exists an integer $p \in \mathbf{Z}$ and homotisms $A \to P(p)$ and $A \to Q(-p)$.

(By means of lemma 1 and prop. 1 of A, X, pp. 66 and 62, reduce to the case where $P$ and $Q$ are zero on the right; by means of prop. 1 of loc. cit. and Theorem 3 of II, § 5, No. 4, construct then complexes $P'$ and $Q'$ such that $P = A \oplus P'$ and $Q = A \oplus Q'$, and prove that one has $H(P') = H(Q') = 0 .$)

b) Let $I, J$ be dualizing $A$-complexes (Exercise 10). If $A$ is local, prove that there exists an integer $n$ and a homotism of $J$ onto $I(n)$ (let $P = \mathrm{Hom}_{\mathbf{A}}(I, J) , Q = \mathrm{Hom}_{\mathbf{A}}(J, I)$; using the morphism $\nu$ of No. 7 and the homotism $\alpha_J : J \to \mathrm{Hom}_{\mathbf{A}}(Q, I)$ relative to the dualizing complex $I$, construct a homotism $P \otimes_A Q \to A$, and apply a)).

c) In the general case, prove that there exists an integer $n$, an $A$-module $L$ projective of rank 1, and a homotism of $J$ onto $I \otimes_A L(n)$ (put $L = H(\mathrm{Hom}_{\mathbf{A}}(I, J))$, and apply b)).