Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful?


If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full subcategory of $D(\operatorname{QCoh}(X))$ consisting of those complexes whose cohomology sheaves are coherent. The bounded above version of the latter statement is here: <https://stacks.math.columbia.edu/tag/0FDA>


Are any counterexamples known? If yes, how does one "usually" define the unbounded derived category of coherent sheaves?


P.S. It appears that the "modification" of $D(\operatorname{Coh}(X))$ consisting of those quasi-coherent complexes whose cohomology sheaves are coherent is more "useful". This is the version that is relevant for Grothendieck duality; isn't it?



Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?
No, not always.


In


*Positselski, Leonid; Schnürer, Olaf M.*, [**Unbounded derived categories of small and big modules: is the natural functor fully faithful?**](https://doi.org/10.1016/j.jpaa.2021.106722), J. Pure Appl. Algebra 225, No. 11, Article ID 106722, 23 p. (2021). [ZBL1464.18015](https://zbmath.org/?q=an:1464.18015).


the authors show in Theorem 3.1 that the natural functor $D(\operatorname{mod}R)\to D(\operatorname{Mod}R)$ is not fully faithful when $R$ is a quasi-Frobenius ring of infinite global dimension (such as $R=\mathbb{Z}/4\mathbb{Z}$).


They also have some positive results. For example, Corollary 5.12 states that $D(\operatorname{coh}X)\to D(\operatorname{Coh}X)$ is fully faithful if $X$ is a regular Noetherian scheme of finite Krull dimension.


