Corollary

Let $A$ be a noetherian ring, $J$ an ideal of $A$, $M$ a finitely generated $A$-module, $\widehat{A}$ and $\widehat{M}$ the separated completions of $A$ and $M$ for the $J$-adic topology. Then one has $\operatorname{prof}_A(J; M) = \operatorname{prof}_{\widehat{A}}(J\widehat{A}; \widehat{M})$.

In fact, the $A$-module $\widehat{A}$ is flat and the $\widehat{A}$-module $\widehat{M}$ is isomorphic to $\widehat{A} \otimes_A M$ (III, § 3, No. 4, Theorem 3); moreover, every maximal ideal of $A$ containing $J$ belongs to the image of the mapping $\operatorname{Spec}(\widehat{A}) \to \operatorname{Spec}(A)$ (loc. cit., Proposition 8).