Timeline for Is there an analogue of curvature in algebraic geometry?
Current License: CC BY-SA 2.5
16 events
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| May 12, 2013 at 17:22 | comment | added | David Ben-Zvi | Good point! :-) but 1. is it clear that this changes the HH^* side of the isomorphism? I would find that surprising for nice enough categories (eg modules for algebras) 2. I have yet to see an example where the original context (derived/triangulated categories as opposed to their refinements) IS what you're actually interested in/what is more useful for applications, as opposed to being an older and more established, simpler but cruder approximation to the "true" invariant, which might be easier to access in some settings but misleading when it gives the wrong answer, as in this case. | |
| May 12, 2013 at 6:15 | comment | added | Mariano Suárez-Álvarez | Well, in order to get an isomorphism, you changed both the domain and the codomain of the map. The original domain and codomains had certain interest of their own, so the fact that after redefining things you do get an isomorphism may be not as useful as it may appear initially when seem from the point of view of the contexts where the question originally arose :-) | |
| Apr 30, 2013 at 3:11 | comment | added | David Ben-Zvi | Thanks! yes this is certainly a triangulated category failing - it's well known that functors and integral transforms don't match in this setting, and that is reflected in the divergence of self-exts of the identity in the two. Yet another reason to prefer dg or $A_\infty$ or (stable) $\infty$-categories.. | |
| Apr 30, 2013 at 0:43 | comment | added | Mariano Suárez-Álvarez | I put a copy of the paper at mate.dm.uba.ar/~aldoc9/tmp/linckelmann.pdf; this sure come from the enhancement. | |
| Apr 30, 2013 at 0:36 | comment | added | David Ben-Zvi | I can't find that specific paper online - could you elaborate the issue? could be a derived category vs dg enhancement issue (you might want to look at Caldararu's Mukai Pairing paper also for a derived category discussion of the issues) | |
| Apr 30, 2013 at 0:06 | comment | added | Mariano Suárez-Álvarez | I'll have to sit down and go through the definitions. What makes noise to me is the fact mentioned, for example, in Example 3.1 of Markus Linckelmann's «Graded centers and -blocks of finite groups.» | |
| Apr 29, 2013 at 23:44 | comment | added | David Ben-Zvi | It is - more precisely End(Id) (in the dg category of endofunctors) is a complex with $H^i(End(I))= Hom(I,I[n])$ (where the latter Hom takes place in the homotopy category) | |
| Apr 29, 2013 at 23:27 | comment | added | Mariano Suárez-Álvarez | Ah. Your $End(Id)$ is then not the direct sum of the spaces of natural transformations $Id\to T^n$ from $Id$ to translations (with the obvious multiplication), I guess? (I only have $D(R-Mod)$ in mind) | |
| Apr 29, 2013 at 21:03 | comment | added | David Ben-Zvi | @Mariano - A good reference for this is Toen's paper arxiv.org/abs/math/0408337 --- maybe it can be seen more classically, but it follows immediately from the identification of functors with bimodules (or integral / Fourier-Mukai transforms) sending the identity to the diagonal bimodule. This is hard (or false) in a model (or triangulated) category setting but easy and formal in an $\infty$-categorical setting (see e.g. arxiv.org/abs/0805.0157 or Gaitsgory's notes on dg categories math.harvard.edu/~gaitsgde/GL ) | |
| Apr 28, 2013 at 8:17 | comment | added | Mariano Suárez-Álvarez | Here you wrote that $HH^*$ is $End(Id)$ in the derived category: do you have a reference for this fact? | |
| Mar 27, 2011 at 17:56 | vote | accept | Paul Siegel | ||
| Mar 27, 2011 at 17:56 | comment | added | Paul Siegel | I think this answer (and Mariano's) settle my question about Chern-Weil theory in the most satisfying possible way. And from what I have learned since I asked this question long ago, it seems that Joel Fine's answer gives about as much as there is to say about "algebraic curvature bounds". I've learned a lot from these answers - thanks everyone! | |
| Jan 7, 2011 at 16:23 | comment | added | David E Speyer | That's a really nice answer! | |
| Jan 7, 2011 at 15:26 | comment | added | David Ben-Zvi | Yes, at least in the formulation I know (which is in the revised version of my Character Theory paper with Nadler, out in a week or two) - a dualizable module over an associative algebra object in a symmetric monoidal ∞-category has a Chern character in cyclic homology. | |
| Jan 7, 2011 at 11:28 | comment | added | Kevin H. Lin | "reasonable finiteness" = dualizable? | |
| Jan 7, 2011 at 4:11 | history | answered | David Ben-Zvi | CC BY-SA 2.5 |