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

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