Counterexamples about Infinitely Generated Modules over Commutative Rings

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).

1. Nakayama’s Lemma (Krull’s Intersection Theorem)

2. \mathrm{Supp}M =V(\mathrm{ann M}).

3. \mathrm{Hom}_A(M,N)\otimes B= \mathrm{Hom}_B(M\otimes_A B, N\otimes_A B) for B a flat A-module

4. M\otimes _R \hat{R} =\hat{M}

5. Finitely Generated Nil Ideal is Nilpotent

6. Finitely presented flat module is projective

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

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:

Depth and Dimension of the Fibre of a Local Ring Homomorphism

An important information of a homomorphism of local rings (R,\mathfrak{m}) \to (S,\mathfrak{n}) is its fibre S/\mathfrak{n}S. For example it relates the dimensions and depth of R and S. 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].