ScalingStacks

We now verify that these three submodules are stable under $d.$ For $X_{2}$ this is obvious. To see it for $X_{3}$, notice that $dz\in\overline{C}(D_{1}(\ast 11))[-2]\{-2\}$ whenever $z\in\overline{C}(D_{1}(\ast 11))[-2]\{-2\}.$ Moreover, for such a $z,$

$$d\beta(z)=-d_{10}^{(-1)}\beta(z)+[-1]\psi_{4}\beta(z)=-d_{10}^{(-1)}\beta(z)+z=\beta d_{11}^{(-2)}(z)+z$$

(111)

The second equality is implied by $[-1]\psi_{4}\beta=\mbox{Id}.$ Map $\psi_{4}\beta$ is associated to the surface