# Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?

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.

• 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. – pbelmans Mar 12 '17 at 20:19
• @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? – Zhaoting Wei Mar 12 '17 at 20:24
• 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). – Yonatan Harpaz Mar 12 '17 at 20:39
• 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. – Leonid Positselski Mar 12 '17 at 23:44
• – Denis Nardin Mar 13 '17 at 0:35