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

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

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