Timeline for Virtual Lefschetz motive
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 29, 2017 at 10:29 | answer | added | Evgeny Shinder | timeline score: 2 | |
| Jun 10, 2013 at 14:08 | comment | added | Dan Petersen | @THC. I agree with your final comment. | |
| Jun 6, 2013 at 18:33 | comment | added | jmc | That might well be. I have never looked at the Grothendieck ring of $k$-varieties before. Maybe someone else can confirm this… | |
| Jun 6, 2013 at 15:45 | comment | added | THC | Ok, so I guess the idea is that in the Grothendieck ring of $k$-varieties one imagines the class $[\mathbf{A}^1]$ to correspond to the Lefschetz motive $\mathbb{L}$ (and therefore using suggestively the same notation), because there one also has the decomposition $[\mathbf{P}^1] = [\mathbf{A}^1] + [\mathrm{Spec}(k)]$, the latter being the $\mathbf{1}$ in the ring. | |
| Jun 6, 2013 at 12:06 | comment | added | jmc | Maybe it helps if you explain which article you are trying to understand. For an introduction to Chow motives, I recommend "Classical motives" by A.J. Scholl. There is more about the Chow-Künneth decomposition in there as well. | |
| Jun 6, 2013 at 12:04 | comment | added | jmc | The affine line is not projective, hence there is no Chow motive corresponding to it. To say that they are analogous objects, I think there should be more arguments than these two "decompositions". But I am certainly not an expert in the field, so maybe the affine line is a good intuition for the Chow motive $\mathbb{L}$. Currently I really think of it as a twist of a point, and the object representing the 1st Chow group (functor). | |
| Jun 6, 2013 at 9:59 | comment | added | THC | @Johan Commelin: isn't it so that $\mathbb{L} = h^2(\mathbf{P}^1)$ is the part in the decomposition which corresponds to $\mathbf{A}^1$? (``Projective line = point + affine line'' so to speak.) If so, what is the connection between $\mathbb{L} = (\mathrm{Spec}(k),\mathrm{id},-1)$ and the affine line $\mathbf{A}^1 = \mathrm{Spec}(k[X])$? | |
| Jun 5, 2013 at 20:01 | comment | added | Dan Petersen | Right. THC, I think the answer to your question is that you're confusing the Grothendieck ring of varieties and the category of Chow motives. | |
| Jun 5, 2013 at 18:45 | comment | added | jmc | Note that the affine line is not a projective variety. Further $\mathbb{L} = (\text{Spec}(k), \text{id}, -1)$ is the Chow motive that falls out of the canonical decomposition $(\mathbb{P}^{1}, \text{id}, 0) = \mathbb{1} \oplus \mathbb{L}$, where $\mathbb{1} = (\text{Spec}(k), \text{id}, 0)$. This decomposition comes from looking at the graph of projection to a rational point of $\mathbb{P}^{1}$ (and does not depend on the point). Finally, for Chow motives, one starts with smooth projective varieties. | |
| Jun 5, 2013 at 17:01 | history | asked | THC | CC BY-SA 3.0 |