ScalingStacks

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$. There are isomorphisms of groups $R(Z_{1})\times\cdots\times R(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R(Z)$ and $L(Z_{1})\times\cdots\times L(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L(Z)$ given by the inclusions $Z_{i}\hookrightarrow Z$. They induce an isomorphism of groups

(7.3.3)

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