Compact quasi-coherent sheaves Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category. 
 A: Let
$X$ be a  Noether scheme. Then $F$ is a compact object of $Qch(O_X)$ iff $F\in Coh(O_X)$.
Proof learned from a friend: Let $colim_iG_i$ be a filtered colimit in $Qch(O_X)$,
Step1. We prove that $O_X$ is compact: Because the underlying topology of $X$ is locally noether, we have $colim_i\Gamma(X,G_i)\to \Gamma(X,colim_iG_i)$ is an isomorphism by  01FF.
Step2. If $O_X^n\to O_X^m\to F\to0$ is an exact sequence, then there is a commutative diagram with exact rows here.
The two vertical homomorphisms on the right are isomorphisms, so is the vertical one on the left.
Step3. Assume that $F\in Coh(O_X)$, then there is an open cover $\{U_{\alpha}\}_{\alpha}$ of $X$ st for every $\alpha$, there is an exact sequence $O_X^{n_{\alpha}}|_{U_{\alpha}}\to O_X^{m_{\alpha}}|_{U_{\alpha}}\to F|_{U_{\alpha}}$. Since the category $Ab$ satisfies AB5, $colim_iHom(F,G_i)$ is the equalizer of the two natural maps $\prod_{\alpha}colim_iHom(F|_{U_{\alpha}},G_i)\to \prod_{\alpha,\beta}colim_iHom(F|_{U_{\alpha}\cap U_{\beta}},G_i)$. The following commutative diagram has exact rows [see here].
By last paragraph, the natural map $colim_iHom(F|_{U_{\alpha}},G_i|_{U_{\alpha}})\to Hom(F|_{U_{\alpha}},(colim_iG_i)|_{U_{\alpha}})$ is an isomorphism.  Hence, the two vertical homomorphisms on the right are isomorphisms, so is the vertical one on the left. This proves that $F$ is compact.
Step4. Conversely, assume that $F$ is a compact object. By a result of Deligne (Proposition 2, p.407 of Residues and duality by Hartshorne), $F=colim_iF_i$ is a filtrant colimit of a family of coherent modules $\{F_i\}$. Then $Hom(F,F)=Hom(F,colim_i F_i)=colim_iHom(F,F_i)$, so there is $i_0\in I$ and $\iota:F\to F_i$ st $r\iota=Id_F$, where $r:F_i\to F$ is the canonical morphism. Then $r$ is surjective, so $F$ is finite type. For every open subset $U$ of $X$, $n\ge 1$, the kernel of $f:O_X^n|_U\to F|_U$ is the kernel of $\iota f:O_X^n|_U\to F_i|_U$, so finite type. Thus, $F$ is coherent.
