Question

We know that a submodule A of B is pure if and only if the functor $Hom(M, -)$ is exact on the sequence $ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ for every finitely presented module M. So, let X be a scheme and ${\cal G}$ be a subsheaf of ${\cal H}$. Is there any equivalent statement for pure sheaves. ($\cal G$ is pure in $\cal H$, if ${\cal G}(U)$ is pure in ${\cal H}(U)$ for every affine open subset U of X.)

Answer 1

Let $\mathbf{A}$ be a cocomplete abelian category, such as any category of sheaves. An object $M$ is finitely presented if the representable functor $\operatorname{Hom}_{\mathbf{A}}(M,-)$ preserves filtered colimits. Now that you know what a finitely presentable object is, just mimick the definition you know of pure subobject. All this is standard.