Counterexample of $\text{Supp}(M)=\mathbb{V}(\text{Ann}(M))$

Question

This is an exercise from Atiyah-MacDonald: Show that when $M$ is finitely generated $A$-module, we have $\text{Supp}(M)=\mathbb{V}(\text{Ann}(M))$.

Can somebody give counterexamples when $M$ is not finitely generated?

Answer 1

Let $R=k[x]$ and $M=\bigoplus_{t\in k} k[x]/(x-t)$ for an algebraically closed field $k$. Then the support of $M$ clearly contains all the closed points of $\Bbb A^1_k=\operatorname{Spec} R$, but does not contain the generic point, as $M_0=0$. So this support cannot be a closed subset of $\Bbb A^1_k$, which means it's not $V(I)$ for any ideal $I$.

Answer 2

Let $(R, {\frak m} )$ be a Noetherian local ring of positive dimension. Then ${\rm Supp} \, {\rm E}(R/{\frak m}) =\{ {\frak m} \} $, but $\mathbb{V}( {\rm ann} \, {\rm E}(R/{\frak m}) ) = {\rm Spec (R)}$, because ${\rm E}(R/{\frak m})$ is faithful.