Timeline for Let $X$ be a projective variety. Is the bounded derived category of perfect complexes admissable in $D^b(X)$?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2016 at 13:44 | vote | accept | Elle Najt | ||
| Oct 22, 2016 at 11:19 | comment | added | Denis Nardin | @Sasha I am silently using the fact that $D^b(X)$ has a natural enhancement as a stable $\infty$-category. Sorry I didn't think it was worth mentioning (especially since the particular enhancement, you work with does not matter). | |
| Oct 22, 2016 at 4:18 | comment | added | Sasha | @DenisNardin: And how do you define a homotopy colimit over such a category in a triangulated category? All I know is how to define a homotopy colimit over naturals (as a cone of a morphism between infinite direct sum, as in the telescope construction), and my impression was that for more complicated categories one has to use enhancements. | |
| Oct 22, 2016 at 0:39 | comment | added | Denis Nardin | @Sasha It is very silly: it is just the homotopy colimit indexed on the category of perfect complexes mapping to $C$. Since the category of perfect complexes is small, this makes sense. | |
| Oct 21, 2016 at 21:23 | comment | added | Sasha | @DenisNardin: What is the CANONICAL way to represent a bounded complex as a homotopy colimit of perfect complexes? | |
| Oct 21, 2016 at 21:22 | answer | added | Sasha | timeline score: 3 | |
| Oct 21, 2016 at 18:24 | comment | added | Denis Nardin | Perfect complexes are not closed under (homotopy) colimits nor (homotopy) limits, so there can be neither a left nor a right adjoint to the inclusion. The nearest thing I can think of is understanding Db(X) as "things assembled from perfect complexes", since every object is (canonically) a (homotopy) colimit of perfect complexes | |
| Oct 21, 2016 at 18:02 | history | asked | Elle Najt | CC BY-SA 3.0 |