It's well known that under some "weak" hypothesis, such as finitely generated, the support of an A-module is closed in Spec(A). It is true also in the most general case?
Asked
Active
Viewed 709 times
1 Answers
6
Pick a non-closed subset $S$ of $\operatorname{Spec}\mathbb Z$, and let $\displaystyle M=\bigoplus_{\mathfrak p\in S}\:\mathbb Z/{\mathfrak p}$.
user26857
- 52,094
Mariano Suárez-Álvarez
- 135,076
-
Nice but doesn't seem quite correct to me! What about $S={ (0) }$ ? The generic point is non-closed and you get with your formula $M=\mathbb Z$, whose support is the whole of $\mathbb Z$ obviously, hence closed. To correct this $S$ must contain maximal ideals only. – Niels Nov 27 '14 at 10:47