ScalingStacks

Proof. Lemma 7.4.28 shows that $d(f^{\#}(\theta^{\prime}))=f^{\#}(d(\theta^{\prime}))$ for any $\theta^{\prime}$ and that $d(f(\theta))=f(d(\theta))$ if $|f^{-1}f(\theta)|=1$.

Assume $Z=S^{1}$ (unoriented) and consider a finite subset $M$ of $Z$ as in §7.4.3. We use the notations of that section. It follows from Lemma 7.4.19 that the isomorphism $F$ of Proposition 7.4.18 induces an isomorphism of ${\mathbf{F}}_{2}$-linear categories $F:{\mathbf{F}}_{2}[{\mathcal{H}}_{n}]\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}_{M}(Z)$. It follows now from Lemma 7.4.20 that this isomorphism commutes with $d$. In particular, $d$ is a differential on ${\mathcal{S}}_{M}(Z)$. Since this holds for any finite subset $M$ of $Z$, we deduce that $d$ is a differential on ${\mathcal{S}}(Z)$.

Consider now a non-singular connected $Z$ and an injective morphism of curves $f:Z\hookrightarrow S^{1}$. Since $f$ induces a faithful ${\mathbf{F}}_{2}$-linear functor ${\mathcal{S}}(Z)\to{\mathcal{S}}(S^{1})$ commuting with $d$, we deduce that $d$ is a differential on ${\mathcal{S}}(Z)$.

The decomposition (7.4.3) is compatible with $d$, hence $d$ is a differential on ${\mathcal{S}}(Z)$ for any non-singular $Z$.

Consider now a general $Z$ and $q:\hat{Z}\to Z$ its non-singular cover. Since the additive ${\mathbf{F}}_{2}$-linear functor $q^{\#}$ commutes with $d$, it follows that $d$ is a differential on ${\mathcal{S}}(Z)$.

The last statement of the theorem follows from Lemma 7.3.17. ∎