Theorem 7.4.32 (0PBX) · ScalingStacks

0PBX

Theorem 7.4.32. The map $d$ equips ${\mathcal{S}}(Z)$ with a structure of differential $\bar{\Gamma}(Z,Z_{exc}^{+})$ -graded ${\mathbf{F}}_{2}$ -linear category and ${\mathcal{S}}^{\bullet}(Z)$ with a structure of differential $\bar{\Gamma}(Z,Z_{exc}^{+})$ -graded pointed category.

Let $f:Z\to Z^{\prime}$ be a morphism of curves.

$\bullet\$ The functor $f:{\mathcal{S}}_{f}^{\bullet}(Z)\to{\mathcal{S}}^{\bullet}(Z^{\prime})$ is a faithful pointed functor and its restriction to ${\mathcal{S}}_{\{z\in Z\ |\ |f^{-1}f(z)|=1\}}^{\bullet}(Z)$ is a differential $\bar{\Gamma}(Z^{\prime},Z_{exc}^{\prime+})$ -graded pointed functor.

$\bullet\$ If $f$ is strict, then $f^{\#}:\operatorname{add}\nolimits({\mathcal{S}}(Z^{\prime}))\to\operatorname{add}\nolimits({\mathcal{S}}_{f}(Z))$ is a differential $\bar{\Gamma}(Z^{\prime},Z_{exc}^{\prime+})$ -graded functor commuting with coproducts.

$\bullet\$ If $f$ is a quotient morphism, then $f^{\#}$ is faithful and every map in ${\mathcal{S}}^{\bullet}(Z^{\prime})$ is in the image by $f$ of a map of ${\mathcal{S}}^{\bullet}_{f}(Z)$ .

0PBY

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. ∎