ScalingStacks

Proof. Note that

$$i([\gamma_{0}],\operatorname{id}\nolimits_{z})\leq\min_{\begin{subarray}{c}\gamma\text{ admiss.}\\ [\gamma]=[\gamma_{0}]\end{subarray}}i(\gamma,\operatorname{id}\nolimits_{z})\leq i(\gamma_{0},\operatorname{id}\nolimits_{z}).$$

The third equality of the lemma follows from Lemma 7.1.21.

When $Z=S^{1}$ unoriented, the lemma follows from Lemma 6.2.3.

When $Z$ is a connected non-singular curve, there is an injective morphism of curves $f:Z\to S^{1}$. We have $i([\gamma_{0}],\operatorname{id}\nolimits_{z})\geq i(f([\gamma_{0}]),\operatorname{id}\nolimits_{f(z)})=i(f(\gamma_{0}),\operatorname{id}\nolimits_{f(z)})=i(\gamma_{0},\operatorname{id}\nolimits_{z})$, hence the first two equalities of the lemma hold for $Z$. It follows that they hold for any non-singular curve.

Consider now a general $Z$ and let $q:\hat{Z}\to Z$ be the non-singular cover. Let $\hat{\gamma}_{0}$ be the lift of $\gamma_{0}$ to $\hat{Z}$. We have

$$i(\gamma_{0},\operatorname{id}\nolimits_{z})=\sum_{\hat{z}\in f^{-1}(z)}i(\hat{\gamma}_{0},\operatorname{id}\nolimits_{\hat{z}})=\sum_{\hat{z}\in f^{-1}(z)}([\hat{\gamma}_{0}],\operatorname{id}\nolimits_{\hat{z}})\leq i([\gamma_{0}],\operatorname{id}\nolimits_{z}).$$

We deduce that the first two equalities of the lemma hold.

The last equality of the lemma follows from (7.3.2). ∎