Proposition 4

Let $\rho : A \to B$ be a ring homomorphism, $J$ an ideal of $A$ and $N$ a $B$-module. One has the equality $\operatorname{prof}_A(J; N) = \operatorname{prof}_B(JB; N)$.

Let $x = (x_i)_{i \in I}$ be a generating family of $J$; the family $\rho(x) = (\rho(x_i))_{i \in I}$ generates $JB$. By construction the complex $K^\bullet(\rho(x), N)$ is equal to $K^\bullet(x, N)$. The proposition therefore follows from Theorem 1.