Timeline for Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 13, 2017 at 19:57 | comment | added | Hacon | Sometimes, starting from a very geometric situation one has to take homotopy limits and colimits which are no longer in $D_{coh}(X)$. A natural example of this is when considering Cartier modules. See Thm. 1.3.1 of arxiv.org/pdf/1310.2996.pdf for an example with nice geometric consequences where I do not know how to prove the result in $D_{coh}(X)$. | |
| Mar 13, 2017 at 0:35 | comment | added | Denis Nardin | Remember, it is better to have a good category with bad objects, than a bad category with good objects. | |
| Mar 12, 2017 at 23:44 | comment | added | Leonid Positselski | 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. | |
| Mar 12, 2017 at 20:39 | comment | added | Yonatan Harpaz | 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). | |
| Mar 12, 2017 at 20:24 | comment | added | Zhaoting Wei | @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? | |
| Mar 12, 2017 at 20:19 | comment | added | pbelmans | 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. | |
| Mar 12, 2017 at 19:41 | history | asked | Zhaoting Wei | CC BY-SA 3.0 |