A question of ATIYAH-MACDONALD commutative algebra book:
In question nº 19 vii), given on the page 46 it is stated that –
For a finitely generated $A$-module $M$ where $A$ is a commuative ring with identity and for an ideal $J$ of $A$, the support of the quotient module $M/JM$ is the same as the set of all prime ideals of $A$ containing the ideal $J+Ann(M)$.
My question is, is it necessary to assume that $M$ is finitely generated $A$-module i.e. does the result true without the assumption finitely generated.
The support of a $A$-module $M$ is defined as set of all prime ideals $\mathfrak{p}$ such that localization of $M$ at $\mathfrak{p}$ is non-zero.
Best Answer
The answer is ‘yes, it is necessary’.
Hint:
Take $J=0$. The assertion would mean that $\operatorname{Supp}M=V(\operatorname{Ann}_AM)$, even if $M$ is not finitely generated.
However, in Bourbaki, Commutative Algebra, ch. II, Localisation, §4, exercise 22 c), you have the following counter-example: let $p$ be a prime number and set $$M=\bigoplus_{k\in\mathbf N}\mathbf Z/p^k\mathbf Z.$$ Then $\operatorname{Supp}M$ is closed, but distinct from $V(\operatorname{Ann}_{\mathbf Z}M)$.