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 !
 A: 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 $K_0(\text{Var})\to K_0(\text{Mot})$
