What is the motivic cohomology $H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ of projective space? By the projective bundle formula, one has


$H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ =
$\oplus\_{i=0}^n\mathrm{Hom}\_\mathbf{DM}(\mathbf{Z}(i)[2i],\mathbf{Z}(q)[p])$


=
$\oplus\_{i=0}^n\mathrm{Hom}\_\mathbf{DM}(\mathbf{Z}, \mathbf{Z}(q-i)[p-2i])$


Is this equal to $\mathbf{Z}$ for (p,q) = (i,2i), $i=0, \ldots, n$ and $0$ else?



The motivic cohomology of projective space
Motivic cohomology is an absolute invariant not a geometric one. The projective bundle formula is purely geometric. It reduces the computation of the motivic cohomology of the projective space to that of the base: 
$$
 H^{p,q}(\mathbb{P}^n\_k) = \bigoplus\_{i=0}^n H^{p-2i,q-i}(Spec(k))
$$
In general $H^{\bullet,\bullet}(Spec(k))$ is highly non trivial. For example, $H^{1,1}(Spec(k)) = \mathbb{H}^1(Spec(k),\mathbb{G}\_m[-1]) = k^\times$ so you will always have $H^{1,1}(\mathbb{P}^n\_k) \neq 0$ (except for $k = \mathbb{F}\_2$).


