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Hi there,

I have a question which popped up while reading papers on motives.

Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then $\mathbb{L}$ is the class $[\mathbf{A}^1]$. I read in several places that in the Grothendieck ring of motives of $V_k$, $\mathbb{L}$ corresponds to the class $[(\mathrm{Spec}(k),\mathrm{id},-1)]$. Why is that? (Where is the affine line gone suddenly?)

Thanks so much !

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  • $\begingroup$ 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. $\endgroup$
    – jmc
    Commented Jun 5, 2013 at 18:45
  • $\begingroup$ 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. $\endgroup$
    – Dan Petersen
    Commented Jun 5, 2013 at 20:01
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    $\begingroup$ 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. $\endgroup$
    – jmc
    Commented Jun 6, 2013 at 12:06
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    $\begingroup$ 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. $\endgroup$
    – THC
    Commented Jun 6, 2013 at 15:45
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    $\begingroup$ @THC. I agree with your final comment. $\endgroup$
    – Dan Petersen
    Commented Jun 10, 2013 at 14:08

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In characteristic zero, there is a canonical ring homomorphism from the Grothendieck ring of varieties to the Grothendieck ring of the additive tensor category of Chow motives (and the latter ring coincides with the Grothendieck ring of the triangulated category of Voevovsky's motives by a result of Bondarko).

This "motivic realization" homomorphism $K_0(Var/k) \to K_0(CHM(k))$ sends the class $[X]$ of a smooth projective variety to the class of its motive $[M(X)]$. The fact that this is well-defined follows from Bittner's presentation of the Grothendieck ring varieties via smooth projective varieties and smooth blow ups, which relies on the Weak Factorization Theorem, hence the characteristic zero assumption. Note that the homomorphism above was constructed much before Weak Factorization by Gillet and Soule (in characteristic zero).

Substituting $\mathbf P^1$ into this map results in that the image of $\mathbf L$ is the Tate motive $\mathbf{L}$ (beware that depending on conventions it may be the dual of the Tate motive here).

Some properties of the motivic realization are discussed here: Some questions about the map $K_0(\text{Var})\to K_0(\text{Mot})$

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