Timeline for Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 30, 2021 at 20:24 | vote | accept | Sofía Marlasca Aparicio | ||
| Apr 30, 2021 at 20:24 | comment | added | Sofía Marlasca Aparicio | @DustinClausen thanks a lot! I probably just need to make myself more familiar with Lurie's work to truly appreciate this... | |
| Apr 23, 2021 at 12:29 | answer | added | Peter Scholze | timeline score: 6 | |
| Apr 9, 2021 at 18:40 | comment | added | Dustin Clausen | Sorry if this all just sounds like a technical jumble. Let me know what further explanations you want. | |
| Apr 9, 2021 at 18:39 | comment | added | Dustin Clausen | As to why it holds in this example, the point is the simple structure of the site of extremally disconnected compact Hausdorff spaces: the finite disjoint union topology. This means that evaluation on extr. disconnecteds detects isomoprhism and is t-exact. Thus homology commutes with evaluation on extr. disconnecteds which implies hypercompleteness | |
| Apr 9, 2021 at 18:33 | comment | added | Dustin Clausen | I guess another, more positive, way of saying things is that this is always true if you replace "sheaves with values in D(Ab)" by "hypersheaves with values in D(Ab)". The basic point is that it's only in the hypersheaf context that you can check isomorphisms on homology groups sheaves, which matches up with the equivalences you see in the derived category of sheaves. | |
| Apr 9, 2021 at 18:31 | comment | added | Dustin Clausen | This holds when the site is hypercomplete (or, just every sheaf of HZ-modules is hypercomplete), but not in general. It looks like this is all explained in Marc's answer to your linked question; could you clarify what more you're looking for? | |
| Apr 9, 2021 at 16:33 | history | asked | Sofía Marlasca Aparicio | CC BY-SA 4.0 |