The support of a finite type module on an algebraic space

I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.

In here the schematic support of F is defined by passing to étale covers and reducing to the case of schemes. However the definition I like is via the canonical morphism $O_X \to \underline{End}(F)$. The kernel of this morphism is an ideal sheaf I. I'd like to say that the schematic support of $F$ is the closed algebraic subspace of $X$ defined by I.

EDIT: As mentioned in the comments below, the endomorphism sheaf is not in general quasi-coherent, unless $F$ is of finite presentation. I'm happy to assume $F$ to be of finite type, but I don't want to assume finite presentation. In the general case: could one take the coherator of $\underline{End}(F)$? Does that construction make sense and would that give the right schematic support?

Is it true?

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Do you know how to prove that this definition "works" for schemes, using the Zariski topology? (Even on an affine scheme the sheaf-Hom among quasi-coherent sheaves need not be quasi-coherent, let alone the sheaf associated to the "expected" Hom-module, unless one imposes a finite presentation hypothesis; finite type is insufficient.) If "no" then figure that out before contemplating the case of algebraic spaces. And if "yes" then please explain why you can't apply the same argument to answer your question for algebraic spaces. – user29720 Jun 13 '13 at 10:23
"And if "yes" then please explain why you can't apply the same argument to answer your question for algebraic spaces." - I think you forgot to write wink wink at the end :). but maybe I can ask a real question: when $F$ is only of finite type, can one not use the coherator of the sheaf hom to get a quasi-coherent sheaf? would that do the trick? – Jacob Bell Jun 13 '13 at 15:47
@Jacob: I don't know what a "coherator" is (and suspect it doesn't matter), and the final question in my initial comment was not meant as a joke. If you know the trick to prove in the scheme case that the kernel of $O_X\rightarrow \mathcal{End}(F)$ is quasi-coherent (even though the target is generally not) then you should see it works the same for algebraic spaces, thereby answering your own question. So please tell us what approach you have tried (first for schemes) and where you got stuck, so one can know what advice would be helpful to you. – user29720 Jun 14 '13 at 6:38
I think I understand how you'd prove it, I was just surprised not to find in the stacks project. But then again it's not even in the schemes section, so it must have been exactly because of the reason you mentioned: the sheaf is not finitely presented. – Jacob Bell Jun 14 '13 at 9:58
The coherator is the maximal quasi-coherent sheaf contained in the given sheaf. It might be a substitute for End when the sheaf is only of finite type, but I agree that it doesn't look like it would give you the right support. I thought to ask as you might have known. – Jacob Bell Jun 14 '13 at 9:59