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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?

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up vote 8 down vote accepted

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 $k = \mathbb{F}_2$).

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