ScalingStacks

We define an endomorphism $\tau$ of $E^{2}$ as follows. We have $E^{2}(m,\pi_{21},\pi_{31},\pi_{32})=(m^{\prime\prime},\pi^{\prime\prime}_{21},\pi^{\prime\prime}_{31},\pi^{\prime\prime}_{32})$ where (ignoring differentials)

$$m^{\prime\prime}=E_{3}^{2}(m)\oplus E_{3}E_{2}(m)\oplus E_{3}E_{1}(m)\oplus E_{2}E_{3}(m)\oplus E_{2}^{2}(m)\oplus E_{2}E_{1}(m)\oplus E_{1}E_{3}(m)\oplus E_{1}E_{2}(m)\oplus E_{1}^{2}(m).$$

We define the endomorphism $\tau$ of $m^{\prime\prime}$ by

(4.3.6)

$$\tau=\left(\begin{matrix}\tau_{3}\\ &&&\sigma_{23}\\ &&&&&&\sigma_{13}\\ \\ &&&&\tau_{2}\\ &&&&&&&\sigma_{12}\\ \\ \\ &&&&&&&&\tau_{1}\end{matrix}\right).$$