Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.)

share|improve this question
    
Perhaps you should assume $X$ to be qs qc to get the affine criterion out of a more general one. –  Martin Brandenburg May 25 '11 at 10:47
    
I don't know what you mean by qs qc scheme. May you explain me more details, please? –  Gholam May 28 '11 at 11:04
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.