ScalingStacks

Therefore, in this case the class $\eta$ is redundant (being determined by $\alpha$), so we simply drop it from the notation, writing for example $\alpha$ instead of $(\alpha,\eta)$ for the classes in $H_{2}^{L}(W_{2};{\mathbb{Z}})$. With this in mind, the isomorphism from Theorem 3.2 is written as

(10)

$$\Phi:\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)\xrightarrow{\phantom{a}\cong\phantom{a}}\mathcal{S}_{0}^{N}(W_{2};L,\alpha).$$