ScalingStacks

Proof. The first statement follows from the fact that $F_{I}$ preserves lengths (cf the discussion before Lemma 6.2.2).

Note that $\llbracket s_{i,j}\rrbracket=0$ for $i,j\in\tilde{I}$ with $i-j{\not\in}n{\mathbf{Z}}$, while $\llbracket F_{I}(c)\rrbracket=\delta$. We deduce that $\llbracket\sigma\rrbracket=m\cdot\delta$.

The last statement of the lemma is immediate. ∎