0MXV Proposition 3.4. Let XX be a variety that admits an affine paving. Then the motive with compact support Mc(X)∈DTM(k,ℚ)w=0\operatorname{M}^{c}(X)\in\operatorname{DTM}(k,\mathbb{Q})^{w=0} is pure Tate.
0MXW Proof. This follows from the localisation sequence and Mc(𝔸n)=ℚ(n)[2n].\operatorname{M}^{c}(\mathbb{A}^{n})=\mathbb{Q}(n)[2n]. A proof can be found in [Ebe21, Lemma 2.3(10)]. ∎