Remark 2

For every ideal of definition $\alpha$ of $A$, the $A$-module $\mathrm{Ext}_A(A/\alpha, M)$ is annihilated by $\alpha$, and $A/\alpha$ identifies with $\widehat{A}/\alpha \widehat{A}$; consequently, the graded $A$-module $\mathrm{Ext}_A(A/\alpha, M)$ identifies with $\widehat{A} \otimes_A \mathrm{Ext}_A(A/\alpha, M)$, hence also with $\mathrm{Ext}_{\widehat{A}}(\widehat{A}/\alpha \widehat{A}, \widehat{A} \otimes_A M)$ (A, X, p. 111, prop. 10). The set of ideals $\alpha \widehat{A}$, for $\alpha \in \mathscr{D}$, contains the powers of $m_{\widehat{A}}$, hence is cofinal in the set of ideals of definition of $\widehat{A}$; we therefore deduce from the foregoing a canonical isomorphism of graded $A$-modules

If the $A$-module $M$ is of finite type, the $A$-module $\widehat{A} \otimes_A M$ identifies with the completion $\widehat{M}$ of $M$ (III, § 3, No. 4, th. 3), and we have an isomorphism $H_A(M) \to H_{\widehat{A}}(\widehat{M})$, graded of degree 0.