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?

finite presentationhypothesis; 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 10:23wink winkat 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