Timeline for The derived category of the heart of a t-structure
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 26 at 22:17 | comment | added | R. van Dobben de Bruyn | For reference, there is at least one place in the literature where the exercise from Gelfand–Manin is worked out, namely Lemma 3.3 in the paper Derived categories of hearts on Kuznetsov components by Li, Pertusi, and Zhao. There might be other places. | |
| Jun 10, 2021 at 7:48 | comment | added | AT0 | @Sasha I checked Gelfand-Manin and it's there given as an exercise, thanks! | |
| Jun 10, 2021 at 7:26 | comment | added | Sasha | @AT0: No, sorry, I don't remember this (I have read this 20 years ago). It could be in one of the versions of Gelfand-Manin, or maybe in the Beilinson's paper. | |
| Jun 9, 2021 at 13:02 | comment | added | AT0 | @Sasha Do you recall a reference for the criterion of the functor being an equivalence? | |
| Dec 12, 2016 at 22:52 | comment | added | ACL | Neeman (1991) has a modification of the definition of a triangulated category, where there always exists a functor from the (bounded) derived category of the heart to the ambient triangulated category. | |
| Nov 15, 2012 at 14:59 | comment | added | Akhil Mathew | (Well, at least if the associated abelian category has enough projectives.) | |
| Nov 15, 2012 at 14:58 | comment | added | Akhil Mathew | I just want to point out that if the triangulated category comes from a stable $\infty$-category, then you can produce a natural functor, using essentially the universal property of the derived category (in the $\infty$-categorical context). This is explained in chapter 1 of Lurie's "Higher Algebra." For instance, you can use this property to define the generalized Eilenberg-MacLane spectrum functor (from the derived category of abelian groups to spectra) in Dustin's answer. | |
| Nov 15, 2012 at 11:43 | comment | added | Leo Alonso | Thank you both. In fact, I was wondering about an unbounded version of the realization functor. I'll check Keller's approach. | |
| Nov 15, 2012 at 10:47 | comment | added | Sasha | Definitely, Denis-Charles is the person to ask! I just know that this is one of the ways to use them. | |
| Nov 15, 2012 at 10:34 | comment | added | D.-C. Cisinski | @Leo Alonso: to my knowledge, there is, unfortunately, no published reference for the derivator point of view. However, in some sense which can be made precise, this is just a reformulation of the results in Bernhard Keller's paper "Derived categories and universal problems", Communications in Algebra 19 (1991), 699-747. This is because one can prove that any triangulated derivator defines a tower of triangulated categories in the sense of loc. cit. (or one can directly translate Keller's proof in the language of derivators), which is a nice exercise. | |
| Nov 15, 2012 at 9:45 | comment | added | Leo Alonso | @Sasha Can you give a reference for an approach of the realization functor via deviators? TIA. | |
| Nov 15, 2012 at 5:20 | history | answered | Sasha | CC BY-SA 3.0 |