Exercise 6, § 1

Let $A$ be a local noetherian ring, $P$ a complex of flat $A$-modules, of finite length $\ell$, such that $\mathrm{Supp}(H(P)) = \{m_A\}$. Prove that every $A$-module $M$ such that $\kappa_A \otimes_A M \neq 0$ is of depth $\leq \ell$ (apply prop. 3 to the complex $P \otimes_A M$, observing that this complex is not exact and using exerc. 4).