I was doing Professor Vakil's FOAG, in exercise 9.1M which defines an scheme- theoretic support of a finite generated $A$- module $M$ to be:
the scheme-theoretic intersection of all closed subschemes $\text{Spec } A/I \to \text{Spec } A$ for which $M$ is an $A/I$ module.Show that it is the scheme theoretic union of scheme theoretic supports of any finite generating set of $M$.
My attempt: since $M$ is $A/I$ module iff $I \subset \text{Ann}(M)$. therefore the intersection should be $\text{Spec } A/(\text{Ann }M)$. But why bother to consider the intersection of those $\text{Spec } A/I$ instead of defining it directly to be $\text{Spec } A/(\text{Ann }M)$?