Corollary

Let $A$ be a local ring, $a$ an ideal of $A$ distinct from $A$ and $M$ an $A$-module annihilated by $a$. One has $\operatorname{prof}_A(M) = \operatorname{prof}_{A/a}(M)$.

Let $\rho : A \to B$ be a ring homomorphism, $x = (x_i)_{i \in I}$ a finite family of elements of $A$ and $M$ an $A$-module. For every integer $p$, let us denote by $u^p : B \otimes_A C_I^p(M) \to C_I^p(B \otimes_A M)$ the $B$-linear homomorphism which associates to $b \otimes m$ the alternating mapping $(\alpha_1, \ldots, \alpha_p) \mapsto b \otimes m(\alpha_1, \ldots, \alpha_p)$. The family $(u^p)$ defines an isomorphism of complexes

Let us consider the canonical homomorphism

(A, X, p. 62); by composition with $H^p(u)$, we deduce from it a homomorphism