This post is about examples of some nice things that could fail for modules that are not finitely generated (or not finitely presented when the ring is not Noetherian).
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
An important information of a homomorphism of local rings is its fibre . For example it relates the dimensions and depth of and . We introduce two main theorems that relate the depth and dimensions and some corollaries. Finally we apply them to show that the polynomial ring or formal power series ring over a Cohen-Macaulay ring is Cohen-Macaulay. See also Theorem A.11 in [Bruns, Herzog] and Theorem 23.2 in [Matsumura].