Proposition 4.4 (0MYS) · ScalingStacks

Proposition 4.4. Let $d_{j}=\dim\widetilde{\mathcal{N}}_{j}.$ There is a natural isomorphism

\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}(\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])=\operatorname{CH}_{d_{j}-n}^{G}(Z_{i,j})_{\mathbb{Q}}.

0MYT

Proof. Let $Z=Z_{i,j}$ with projections $\pi_{i,j}:Z\to\widetilde{\mathcal{N}}_{i,j}.$ For a variety $X$ denote by $\operatorname{fin}_{X}:X\to\operatorname{pt}$ the structure map. Then by using various adjunctions, base change and $\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}=\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}(-d_{j})[-2d_{j}]$ since $\widetilde{\mathcal{N}}_{j}$ is smooth, we get

	$\displaystyle\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})}($	$\displaystyle\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}}),\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}},\mu_{i}^{!}\mu_{j,*}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(\widetilde{\mathcal{N}}_{i},\mathbb{Q})}(\operatorname{fin}_{\widetilde{\mathcal{N}}_{i}}^{}\mathbb{Q},\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{*}\mathbb{Q}(n)[2n])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{\widetilde{\mathcal{N}}_{i},}\pi_{i,}\pi_{j}^{!}\operatorname{fin}_{\widetilde{\mathcal{N}}_{j}}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{fin}_{Z,*}\operatorname{fin}_{Z}^{!}\mathbb{Q}(n-d_{j})[2(n-d_{j})])$
		$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\mathbb{Q},\operatorname{M}^{c}(Z)(n-d_{j})[2(n-d_{j})]).$

Now the last term is isomorphic to $\operatorname{CH}^{G}_{d_{j}-n}(Z)_{\mathbb{Q}}$ by [Kel17, Theorem 5.3.14]. ∎