Exercise 4, § 1

Let $A$ be a ring, $J$ a finite type ideal of $A$, $M$ an $A$-module, $P$ a flat $A$-module. Prove that one has $\mathrm{prof}_A(J; P \otimes_A M) \geq \mathrm{prof}_A(J; M)$, and that there is equality if $P$ is faithfully flat.