Timeline for Is there triangulated category version of Barr-Beck's theorem?
Current License: CC BY-SA 2.5
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| Mar 23, 2011 at 22:10 | comment | added | Paul Balmer | Just before Prop. 3.4, Rosenberg says on p.36, that given a monad F on a triangulated category, the category of F-modules is triangulated in such a way that the forgetful functor is exact. This would imply in particular that modules over ring spectra up to homotopy form a triangulated category. Etc, etc. Do you believe this? Specifically, I don't see how to put a module structure on the cone of a morphism of modules, unless some assumption is made about the monad (like being separable, in which case it works). | |
| Mar 23, 2010 at 16:12 | comment | added | Zoran Skoda | Yes, in the usual geometric situations with qcoh sheaves on open neighborhoods the descent for derived categories does not work as we know from elementary counterexamples. But there are lots of examples with Cohn localization which is non-flat, and these are examples in which if we would restrict to noncommutative subvarieties which are close to the commutative the descent would suddenly become flat, and everything would work at the abelian level already. You should push Sasha & Maxim to finish their "secret" preprint started in 1999, whose delay was initially due to lack of similar tools. | |
| Mar 23, 2010 at 16:11 | comment | added | Zoran Skoda | David, I corrected the link, for some reason MPI changed the format of the URL in the meantime and www at the beginning is now mandatory. | |
| Mar 23, 2010 at 16:05 | history | edited | Zoran Skoda | CC BY-SA 2.5 |
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| Mar 18, 2010 at 8:00 | comment | added | Shizhuo Zhang | @David and Skoda: Now I saw Rosenberg used this triangulated version of Beck's theorem for induction theorem in triangulated category. Because in triangulated version, we do not require the exactness of functors,so the induction theorem works very smooth in this settings. For induction theorem, I mean cohomological induction. Rosenberg then used t-structures to turn this triangulated picture to abelian picture. I will elaborate this in an additional answers | |
| Mar 9, 2010 at 13:01 | comment | added | David Ben-Zvi | Thanks Zoran! The link to Rosenberg's pdf didn't work for me. I'm very curious to see how the triangulated theorem is used - I understand how you'd use the $A_\infty$ version (which is presumably subsumed now by Lurie) but I'm wondering if there are some settings where the triangulated version is useful eg for gluing. | |
| Mar 8, 2010 at 19:52 | history | edited | Zoran Skoda | CC BY-SA 2.5 |
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| Mar 8, 2010 at 19:24 | history | answered | Zoran Skoda | CC BY-SA 2.5 |