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Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.

$D^b_{coh}(X)$ is closely related to the geometry of $X$ and has been intensively studied, see for example Huybrechts's book. On the other hand, $D_{qcoh}(X)$ has also been studied but its categorical structure is more complicated. For example, the injective resolutions in $D_{qcoh}(X)$ are actually h-injective complexes and more difficult to construct than the injective resolutions $D^b_{coh}(X)$.

My question is: is there any place that $D_{qcoh}(X)$ rather than $D^b_{coh}(X)$ plays an essential role? By "essential" I mean that $D_{qcoh}(X)$ is more than just an auxiliary tool.

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    $\begingroup$ Certain functors do not necessarily restrict to the bounded derived category. That's a pretty important reason to consider the unbounded derived category if you ask me. $\endgroup$
    – pbelmans
    Commented Mar 12, 2017 at 20:19
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    $\begingroup$ @pbelmans Among the Grothendieck's six functors, it seems that the pushforward does not exist in the bounded derived category of coherent sheaves. Is that what you mean? $\endgroup$
    – Zhaoting Wei
    Commented Mar 12, 2017 at 20:24
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    $\begingroup$ The underlying $\infty$-category of $D_{qcoh}(X)$ is presentable, but the underlying $\infty$-category of $D_{qcoh}^b(X)$ is not (it doesn't have all limits and colimits). $\endgroup$
    – Yonatan Harpaz
    Commented Mar 12, 2017 at 20:39
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    $\begingroup$ The push-forward may take complexes of coherent sheaves to complexes of quasi-coherent sheaves. The pull-back may take bounded complexes to unbounded ones. $\endgroup$
    – Leonid Positselski
    Commented Mar 12, 2017 at 23:44
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    $\begingroup$ Remember, it is better to have a good category with bad objects, than a bad category with good objects. $\endgroup$
    – Denis Nardin
    Commented Mar 13, 2017 at 0:35

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