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 !

smoothprojective varieties. – jmc Jun 5 '13 at 18:45