# Virtual Lefschetz motive

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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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. –  jmc Jun 5 '13 at 18:45
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. –  Dan Petersen Jun 5 '13 at 20:01
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. –  jmc Jun 6 '13 at 12:06
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. –  THC Jun 6 '13 at 15:45
@THC. I agree with your final comment. –  Dan Petersen Jun 10 '13 at 14:08