Associated Points and Applications

Proof of proposition. We know quite generally that Supp s is a closed set (indeed, for any sheaf of abelian groups on any topological space; see Exercise II.1.14 of [1]). On a Noetherian scheme, it is then a finite union of irreducible components, and we simply need to show that each generic point of each component is an associated point of F . But this is clear, since if x is such a generic point, locally around x we have Supp s = x̄, so s annihilates any element of mx ⊆ OX,x. The converse is a bit subtler. We begin by showing that if x ∈ X is an associated point of F , then there is a section s ∈ Fx such that Ann(s) = mx; i.e., fs = 0 if and only if f ∈ mx. We are given that for any f ∈ mx, multiplication by f is not