A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

Question

Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories $$ D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X)) $$ where $D^b(coh(X))$ is the derived category of bounded complexes of coherent sheaves, and $D^b_{coh}(Qcoh(X))$ is the derived category of bounded complexes of quasi-coherent sheaves with coherent cohomologies.

However, the proof of Proposition 3.5 only shows that the functor is essentially surjective. We know that a priori there are more morphisms in $D^b_{coh}(Qcoh(X))$ than in $D^b(coh(X))$. Then how to show that the morphism sets are the same?

Answer 1

Indeed, the proof in the book shows that if $G$ is a bounded complex of quasi-coherent sheafs with coherent cohomologies, then there is a subcomplex $G_1\subseteq G$ with same cohomologies of $G$, where $G_1$ is bounded complex of coherent sheafs. Then you can prove directly that two bounded categories have same hom set.

In fact, here $K^b(Coh(X))$ is cofinal subcategoy of $K^b(Qcoh X)$. In general, the following is true:

Let $S$ is a multiplicatively closed subset of an additive category $\mathcal A$, suppose $\mathcal B$ is full subcategory of $\mathcal A$, if $S\cap \mathcal B$ is also a multiplicatively closed subset of $\mathcal B$. If for any $s: X\rightarrow Y$ in $S$, where $X\in \mathcal A$ and $Y\in \mathcal B$, there exists $t:Z\rightarrow X$ with $Z\in \mathcal B$ such that $st\in S$, then $(S\cap \mathcal B)^{-1}\mathcal B\rightarrow S^{-1}\mathcal A$ is fully faithful.