Let $X$ be a quasi-compact semi-separated scheme and

$$\varepsilon: 0\to A \to B \to C \to 0$$ be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact sequence if $$ 0\to {\rm Hom}(P, A)\to {\rm Hom}(P, B) \to {\rm Hom}(P, C)\to 0$$ is exact for every finitely presented quasi-coherent sheaf $P$. A quasi-coherent sheaf $C$ is called (categorical) flat if every short exact sequence of the form $\varepsilon$ is categorical pure. $C$ is called geometrical flat if the functor $C\otimes -$ is exact. A quasi-coherent sheaf $A$ is called Fp-injective if every exact sequence of the form $\varepsilon$ is categorical pure.

Let $E$ be an injective cogenerator for the category of quasi-coherent sheaves over $X$, then for each quasi-coherent sheaf $F$ by $F^*$ we mean ${\rm\mathcal{H}om}_{qc}(F, E)$. to see more details see http://arxiv.org/abs/1307.1729.

Question: Is the following statement true.

$F$ is cateorical flat if and only if $F^*$ is Fp-injective.

Remark: Actually one can show that every cateorical flat quasi-coherent sheaf is flat and hence it is not hard to see that if $F$ is flat then $F^*$ is Fp-injective. So the question reduces to following.

Let $F^*$ be an Fp-injective quasi-coherent sheaf, is $F$ a categorical flat quasi-coherent sheaf?


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