finitely generated – Thinking Bear

Let $A$ be a commutative noetherian ring, $m$ an ideal of $A$ and $M$ a finitely generated $A$ -module, then we have an isomorphism $M\otimes_A \hat{A} \to\hat{M}$ where the completion is with respect to the $m$ -adic topology [III. 3.4 Theorem 3, Commutative Algebra, Bourbaki].

Note that when $A$ is not noetherian, $M$ being finitely presented would suffice.

Here are two counterexamples showing that the map above need not be injective or surjective when $M$ is not finitely generated: https://math.stackexchange.com/q/1143557

Tag: finitely generated

Counterexamples that Tensor Product of a Non-finitely Generated R-module M with the Completion of R is not the Completion of M