ScalingStacks

(7.4.2)

$${\mathcal{S}}^{\bullet}(Z_{1})\wedge\cdots\wedge{\mathcal{S}}^{\bullet}(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}^{\bullet}(Z)$$

where the grading on the left hand term is deduced from the the $\bigl(\prod_{i=1}^{r}\Gamma(Z_{i},(Z_{i})_{exc}^{+})\bigr)$-grading via (7.3.3) and an isomorphism of ${\mathbf{F}}_{2}$-linear categories

(7.4.3)

$${\mathcal{S}}(Z_{1})\otimes\cdots\otimes{\mathcal{S}}(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}(Z).$$