ScalingStacks

The convolution of two cycles $\alpha\in\operatorname{CH}^{G}_{d_{j}-n}(Z_{i,j})_{\mathbb{Q}}$ and $\beta\in\operatorname{CH}^{G}_{d_{k}-m}(Z_{j,k})_{\mathbb{Q}}$ is given by the formula

(4.1)

$\displaystyle\alpha\star\beta=p_{*}\delta^{!}(\alpha\times\beta)\in\operatorname{CH}^{G}_{d_{k}-n-m}(Z_{i,k})_{\mathbb{Q}}$

where $\alpha\times\beta$ is the exterior product and

	$\displaystyle\delta$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,j}\times Z_{j,k}\text{ and}$
	$\displaystyle p$	$\displaystyle:\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}\to\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{k}=Z_{i,k}$

are the diagonal and projection maps.