Intuition for the Lefschetz motive (Tate motive)?

Question

Yo! Maybe this question is too dumb for mathoverflow, but I believe no one will pay attention to it in math.stackexchange, so I will post it here. If this question is not suitable, just delete it.

I will denote $W$ the weak equivalences regarding contractibility of $\mathbb{A}^1$ and $(-)^{\#}$ the pseudo-abelian hull.

I have been trying to understand better the Tate motive and its realizations in some cohomologies. However I always fail to understand why the Lefschetz motive is not something homotopically trivial. First, let me clarify my intuition. As I understand the category $\mathbf{Spc} (k)$ (let's focus on pure effective geometric motives for now) is the smallest category containing schemes and its (non-existent in the category of schemes) cellular decompositions. Now the localization by weak equivalences $\mathbf{Spc} (k) [W^{-1}]$ leads to homotopy types of cellular decompositions of schemes and the Hurewicz map linearize everything and lands in $\mathbf{DM}^{eff}(k)$, which is analogous to $\mathbb{Z} Sing : \mathbf{Top} \rightarrow Ch(\mathbf{Ab})$ (actually it's analogous applying the Hurewicz map before the localization to be more precise). If my intuition is wrong, please, correct me.

Now, in the category of effective geometric motives $\mathbf{DM}^{eff}_{gm}= (K^{b} (Cor_{fin}(k))[\{\text{Mayer Vietoris, W}\}^{-1}])^{\#} \hookrightarrow \mathbf{DM}^{eff}(k) =D^{-} (Sh_{Nis})[W^{-1}]$, the Lefzchetz motive $$\mathbb{L} = \widetilde{[\mathbb{P}^1]} = Cone (x_{*})$$ on LHS and $$\mathbb{L} = C_{*} \mathbb{Z}_{tr} (\mathbb{G}_m, 1) [1]$$ on the RHS, where $x \in \mathbb{P}^1 (k)$.

Now in Virtual Lefschetz motive and http://arxiv.org/pdf/0907.4046v2.pdf for instance, it's used the Grothendieck ring of varieties $K_0 (Var_k)$ as a prototype to pure motives and there $[\mathbb{P}^1] = [Spec (k)] + \mathbb{L} = [Spec (k)] + [\mathbb{A}^1]$, hence $\mathbb{L} \cong [\mathbb{A}^1]$. And thinking intuitively on the complex points $\mathbb{P}^1 (\mathbb{C}) = \{x\} \cup \mathbb{C}$.

Another interpretation that I see is as an orientation class (angle form) of the sphere by noticing that $\mathbb{L} = \mathbb{Z} (1)[2]$ and $H^2 (\mathbb{P}^1)$ have one generator. However in étale cohomology it behaves as the orientation class of the circle (and maybe this is my main confusion: Why $H^2 (\mathbb{P}^1) \cong H^1 (\mathbb{G}_m)$ as Galois representations? Maybe seeing the Hodge structures clarify it?)

So my questions are.

1) Is there a map from or to $K_0 (Var_k)$ to $\mathbf{DM}_{gm}^{eff}(k)$ and in this map what makes $\mathbb{L}$ fails to be contractible?

2) Why $\mathbb{L} = \widetilde{[\mathbb{G}_m]}[1]$ in $\mathbf{DM}_{gm}^{eff}(k)$? In other words why the LHS and RHS above are equal under the embedding of geometric motives into motives? This will explain my confusion with the étale realization of the Lefschetz motive.

3)I would like to read additional remarks or point of views about the Tate and Lefschetz motive which clarify your intuitive understanding.

Thanks in advance.

Answer 1

1) No, there is no such map. You can only send both of these classes into the group $K_0(DM_{gm}^{eff})\cong K_0(Chow^{eff})$. The main point (and, possibly, the source of your problems) is that if you want these two maps to be compatible then you should send variety into its motif with compact support! Thus $[A^1]=[P^1]-[pt]\neq 0$! So, the motif of an affine line is "trivial", whereas its motif with compact support is not.

2. Just look at the Mayer-Vietoris triangle $M(G_m)\to M(A^1)\bigoplus M(A^1)\to M(P^1)\to M(G_m)[1]$. Since motives of affine lines are "trivial", the latter morphism yields the isomorphism in question.

3. If you want to understand the Euler characteristics of (the motif with compact support of) a variety $V$ then just count points in $V(F_q)$. If you want to study "ordinary motives", look at Hodge structures, yes.