Completion and tensor product

Question

Let $(A,\mathfrak m)$ be a local ring and let $M$ be an $A$ module. If we denote with $\widehat M$ the completion of $M$ with respect the filtration:

$$\ldots\supset\mathfrak m^{i-1} M\supset\mathfrak m^{i} M\supset\mathfrak m^{i+1} M\supset\ldots$$ Is the following isomorphism true?

$$\widehat{M}\cong M\otimes_A\widehat{A}$$

Where clearly $\widehat{A}$ is the completion of $A$.

Answer 1

This is true if $A$ is noetherian and $M$ is finitely generated (Bourbaki, Commutative Algebra, Ch. 3, Graduations, Filtrations and Topologies, § 3, n°4, Theorem 3).