# What is interesting about the support of a module?

In Commutative Algebra, we defined the Support of a Module $$M$$ $$\operatorname{Supp}(M) = \{P \in \operatorname{Spec}(R) : M_P \neq \{0 \} \} = \{P \in \operatorname{Spec}(R): \exists m \in M: \operatorname{Ann}(m) \subseteq P \}$$ Today, I asked our tutor "Can you tell us why $$\operatorname{Supp}(M)$$ is interesting?" and he told me no, that he has never really seen it in action.

So I want to pose this question here. Why should $$\operatorname{Supp}(M)$$ be interesting?

• For finitely generated modules the dimension of M is the supremum of the lengths of chains of primes in Supp(M). Jun 24 '19 at 19:52

Consider a sheaf $$\mathcal{F}$$ of $$\mathcal{O}_X$$-modules. Then the support of $$\mathcal{F}$$ is all the points of $$X$$ "where the stalk is non-zero", i.e. $$\mathrm{supp}(\mathcal{F}) = \{ x \in X \mid \mathcal{F}_x \neq 0 \}.$$ This indeed coincides with the definition you've given in your question. Note that one can generalise this notion to complexes of sheaves, i.e. $$\mathrm{supp}(\mathcal{F}^\bullet) = \bigcup \mathrm{supp}(H^i(\mathcal{F}^\bullet)).$$ Then this notion is important in stating/proving many results from algebraic geometry. For example, if we know somthing like $$x \notin \mathrm{supp}(\mathcal{F}^\bullet)$$ then we can deduce that $$\mathcal{F}^\bullet |_U$$ is trivial, where $$x \in U \subset X$$ is an open neighbourhood of $$x$$.