Timeline for why are motives more serious than "naive" motives?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 19, 2014 at 16:04 | comment | added | Simon Pepin Lehalleur | @Adeel: the semisimple objects in the abelian category of mixed motives are conjecturally the motives for the numerical equivalence. The relationship with the embedding of Chow in DM is a bit subtle; Chow is not in the heart of the conjectural motivic t-structure, but is the heart of Bondarko's (non-conjectural) weight structure. The weight filtration on MM (with graded pieces in $M_{num}$) should arise from the interaction of the two structures. See e.g. arxiv.org/abs/1105.0420 | |
| Nov 7, 2014 at 4:00 | answer | added | Will Sawin | timeline score: 12 | |
| Nov 7, 2014 at 3:02 | answer | added | Daniel Litt | timeline score: 20 | |
| Nov 6, 2014 at 18:15 | answer | added | Dan Petersen | timeline score: 12 | |
| Nov 6, 2014 at 17:50 | comment | added | Vivek Shende | I wish this question was tripartide: there are also the original Chow motives, and I'd like to know where they fit in visavis the above analogies. | |
| Nov 6, 2014 at 7:34 | comment | added | Donu Arapura | @birk my comment was partly a joke but not completely. If $R$ is an Artinian ring, then from the class of a module in $K_0(R) $ you can recover its length but you've lost everything else. In the same way passing to $K_0(Var)$ kills a lot; I don't see how you would recover the higher Chow group from its class. | |
| Nov 6, 2014 at 1:43 | comment | added | user40276 | You don't want just motivic cohomology, as I understand, you want the triangulated category of motives (actually the abelian, or the $t$-structure to get the conjectural category). And using Dold-Kan correspondence you can work with spectra that is more suitable for (stable) homotopy theory... | |
| Nov 5, 2014 at 20:58 | comment | added | ACL | @DonuArapura. I'd rather say: for the same reason that cohomology groups are more "serious" than the Betti numbers. | |
| Nov 5, 2014 at 19:02 | comment | added | birk | Of course, I agree with you that this is extremely important. Still, if you only want the motivic cohomology of X you can use higher Chow groups or K-theory as your definition... | |
| Nov 5, 2014 at 18:47 | comment | added | Matthias Wendt | Voevodsky's motives are a category, and you can consider morphisms between motives. This is relevant if you want to define motivic cohomology and study regulators defined on those (e.g. for arithmetic applications). | |
| Nov 5, 2014 at 18:26 | comment | added | birk | Could you elaborate a bit more? Why the E-polynomial, which is a motivic measure on $K_0(Var_k)$ is not enough to study the Hodge numbers? | |
| Nov 5, 2014 at 18:18 | comment | added | Donu Arapura | For the same reason that Betti numbers are more "serious" than the Euler characteristic. | |
| Nov 5, 2014 at 17:40 | review | First posts | |||
| Nov 5, 2014 at 18:09 | |||||
| Nov 5, 2014 at 17:39 | history | asked | birk | CC BY-SA 3.0 |