Relationship between a sheaf and the sheaf we get after applying the coherator.

Let XXX be a quasi-compact and
quasi-separated scheme. Now, consider the forgetful functor ii from the category of quasi-coherant sheaves on XX to category of OXO_X-modules.
i:Qcoh(X)↪Mod(X)i: Qcoh(X) \hookrightarrow Mod(X)
The right adjoint to this functor is called the coherator.
Q:Mod(X)→Qcoh(X)Q :Mod(X) \rightarrow Qcoh(X)

My question is if F∈Mod(X)F \in Mod(X) be a sheaf, then is Q(F)Q(F) is a subsheaf of FF in the category Mod(X)Mod(X)? More precisely is it true that,
i∘Q(F)⊂Fi\circ Q(F)\subset F
If not, can it be true with some additional conditions ?

Basically, I am trying to understand the connection between FF and Q(F)Q(F).

Thanks in advance!

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Have you looked at the affine case?
– Ben
Oct 20 at 19:32

  

 

In case X=SpecAX = SpecA is affine, then Q(F)=F(X)∼Q(F) = F(X)^{\sim}. So, what we want is to see if F(X)∼(U)⊂F(U)F(X)^{\sim}(U) \subset F(U) for all U⊂XU \subset X. Now, it is enough to show this for every affine UU. Let U=SpecBU = SpecB. Then, F(X)∼(SpecB)=F(X)⊗BF(X)^{\sim}(SpecB) = F(X) \otimes B. So the question is F(X)∼(SpecB)⊂F(SpecB)F(X)^{\sim}(SpecB) \subset F(SpecB) ?
– Sam
2 days ago

  

 

This doesn’t seem to me to be true in general. However, I am wondering if this can be true with some further conditions.
– Sam
2 days ago

  

 

By looking at stalks: if XX is affine and FF is globally generated, then the morphism i∘Q(F)→Fi\circ Q(F)\to F is an epimorphism. Thus, to produce a counter-example it suffices to construct a non-quasi-coherent, but globally generated OX\mathcal{O}_X-module on an affine scheme.
– Ben
2 days ago

  

 

Or you just look at the answers to this question: math.stackexchange.com/q/467197/12885
– Ben
2 days ago

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