Subsection 8.1.6. (0PI2) · ScalingStacks

8.1.6. Duality

Let $Z^{\prime}={\mathbf{R}}$ be the smooth curve with $Z^{\prime}_{o}=(-\frac{1}{2},\frac{1}{2})$ , with its standard orientation. Consider a morphism of curves $\tilde{\xi}:Z^{\prime}\to Z$ such that $\tilde{\xi}(Z^{\prime})$ is a component of $Z$ .

Fix an increasing homeomorphism $\alpha:{\mathbf{R}}_{>0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{>\frac{1}{2}}$ fixing the positive integers and define $\alpha^{\prime}:{\mathbf{R}}_{<0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{<-\frac{1}{2}}$ by $\alpha^{\prime}(t)=-\alpha(-t)$ . Let $\xi^{+}=\tilde{\xi}\circ\alpha:{\mathbf{R}}_{>0}\to Z$ and $\xi^{-}=\tilde{\xi}\circ\alpha^{\prime}:{\mathbf{R}}_{<0}\to Z$ . These are injective morphisms of curves, $\xi^{+}$ is outgoing for $Z$ and $\xi^{-}$ is incoming for $Z$ .

Given $n\geq 0$ , we denote by $\theta(n)\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(\{-n,\ldots,-1\},\{1,\ldots,n\})$ the braid given by $\theta(n)_{-i}=[-i\to i]$ .

Let $T$ and $T^{\prime}$ two finite subsets of $Z$ and $I\subset{\mathbf{Z}}_{\geq 1}$ finite. Assume that $\tilde{\xi}(-I)\subset T$ and that given $x\in{\mathbf{R}}$ with $x<i$ for all $i\in-I$ , we have $\tilde{\xi}(x){\not\in}T$ . Assume also that $\tilde{\xi}(I)\subset T^{\prime}$ and that given $x\in{\mathbf{R}}$ with $x>i$ for all $i\in I$ , we have $\tilde{\xi}(x){\not\in}T^{\prime}$ .

We consider the pointed map

	$\displaystyle\kappa_{I}:\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T,T^{\prime})$	$\displaystyle\to\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T\setminus(T\cap\tilde{\xi}(-I)),T^{\prime}\setminus(T^{\prime}\cap\tilde{\xi}(I)))$
	$\displaystyle\theta$	$\displaystyle\mapsto\begin{cases}(\theta_{t})_{t\in T\setminus\tilde{\xi}(-I)}&\text{ if }\chi(\theta)(\tilde{\xi}(-i))=\tilde{\xi}(i)\text{ for }i\in I\\ 0&\text{ otherwise.}\end{cases}$

We put $\kappa_{n}=\kappa_{\{1,\ldots,n\}}$ . Note that $\kappa_{n}=\kappa_{\{n\}}\circ\cdots\circ\kappa_{\{2\}}\circ\kappa_{\{1\}}$ .

Let $f:Z\to\bar{Z}$ be a morphism of curves such that $f\circ\tilde{\xi}$ is a homeomorphism from $Z^{\prime}$ to a component of $\bar{Z}$ . Put $\tilde{\bar{\xi}}=f\circ\tilde{\xi}$ . Denote by $\bar{\kappa}_{n}$ the map defined as above with $Z$ replaced by $\bar{Z}$ .

Let $T$ and $T^{\prime}$ be two finite subsets of $Z$ such that $|f(T)|=|T|$ and $|f(T^{\prime})|=|T^{\prime}|$ . Put $\mathring{T}=T\setminus(T\cap\tilde{\xi}(\{-n,\ldots,-1\})$ and $\mathring{T}^{\prime}=T^{\prime}\setminus(T^{\prime}\cap\tilde{\xi}(\{-n,\ldots,-1\})$ . There is a commutative diagram

(8.1.6)

Similarly, if $f$ is strict and $U$ and $U^{\prime}$ are two finite subsets of $\bar{Z}$ , there is a commutative diagram

(8.1.7)

where $T$ (resp. $T^{\prime}$ ) runs over finite subsets of $Z$ such that $f(T)=U$ (resp. $f(U^{\prime})=T^{\prime}$ ).

0PCJ

Lemma 8.1.14. The map $\kappa_{n}$ commutes with differentials.

0PCK

Proof. Assume first $\tilde{\xi}$ is a homeomorphism and $Z_{o}=\emptyset$ . Let $T$ and $T^{\prime}$ be two finite subsets of ${\mathbf{R}}$ with same cardinality $m$ . Let $a:\{1,\ldots,m\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}T$ and $a^{\prime}:\{1,\ldots,m\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}T^{\prime}$ be the increasing bijections. There is an isomorphism of differential modules (Proposition 7.4.33) $\phi:\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(T,T^{\prime})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H_{m}$ : given $\theta\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T,T^{\prime})$ non-zero and given $i\in\{1,\ldots,m\}$ , we put $\phi(\theta)(i)=a^{\prime-1}(\theta_{a(i)}(1))$ .

Assume in addition that $\{-n,\ldots,-1\}\subset T$ and $T\setminus\{-n,\ldots,-1\}\subset(-1,\infty)$ and $\{1,\ldots,n\}\subset T^{\prime}$ and $T^{\prime}\setminus\{1,\ldots,n\}\subset(-\infty,1)$ . There is a commutative diagram

The lemma follows now from §6.1.1.

Assume now $Z$ is smooth. If $Z(\xi^{+})$ is unoriented, then the lemma holds by the discussion above, using §7.4.10. In general, we consider the morphism of curves $f:Z\to\bar{Z}$ that is an isomorphism outside $Z(\xi^{+})$ and the identity on $Z(\xi^{+})$ , with $f(Z(\xi^{+}))_{o}=\emptyset$ . The vertical maps of the commutative diagram (8.1.6) are injective, hence the lemma holds for $Z$ since it holds for $\bar{Z}$ .

Consider now a general $Z$ . Let $f:\hat{Z}\to Z$ be a non-singular cover. The vertical maps of the commutative diagram (8.1.7) are injective, hence the lemma holds for $Z$ since it holds for $\hat{Z}$ . ∎

Let $M$ be a subset of $Z\setminus\tilde{\xi}\bigl((-\infty,-1]\cup[1,\infty)\bigr)$ .

Given $S$ a finite subset of $M$ , the pointed map

L_{\xi^{+}}^{\bullet}(T,S,e^{n})\wedge R_{\xi^{-}}^{\bullet}(S,T^{\prime},e^{n})\to\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T^{\prime},T),\ (\theta^{\prime},\theta)\mapsto\kappa_{n}(\theta^{\prime}\cdot\theta)

induces an ${\mathbf{F}}_{2}$ -linear map

	$\displaystyle\hat{\kappa}(T,S):L_{\xi^{+}}(T,S,e^{n})$	$\displaystyle\to\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)^{\operatorname{opp}\nolimits}\operatorname{\!-diff}\nolimits}(R_{\xi^{-}}(S,-,e^{n}),\operatorname{Hom}\nolimits(-,T))$
	$\displaystyle\theta^{\prime}$	$\displaystyle\mapsto\bigl((\theta\in R_{\xi^{-}}^{\bullet}(S,T^{\prime},e^{n}))\mapsto\kappa_{n}(\theta^{\prime}\cdot\theta)\bigr).$

0PCL

Proposition 8.1.15. The map $\hat{\kappa}$ induces an isomorphism of differential pointed bimodules $L_{\xi^{+}}(-_{2},-_{1},e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R_{\xi^{-}}(-_{1},-_{2},e^{n})^{\vee}$ .

0PCM

Proof. Lemma 8.1.14 shows that $\hat{\kappa}$ commutes with differentials.

Let $S$ be a finite subset of $M$ of cardinality $n$ .

Assume $\tilde{\xi}$ is a homeomorphism and $Z_{o}=\emptyset$ . There is a commutative diagram (see the proof of Lemma 8.1.14 with $(T,T^{\prime})=(S,\xi^{+}(\{1,\ldots,n\})$ and $(T,T^{\prime})=(\xi^{-}(\{-n,\ldots,-1\}),S)$ )

The bottom horizontal map is bijective by Corollary 3.1.2, hence $\hat{\kappa}(\emptyset,S)$ is bijective.

Assume now $Z(\xi^{+})$ is smooth and unoriented. The map $\hat{\kappa}(\emptyset,S)$ is the same for $Z$ and for $Z(\xi^{+})$ , so $\hat{\kappa}(\emptyset,S)$ is still bijective.

Assume $Z(\xi^{+})$ is smooth. There is a morphism of curves $f:Z\to\bar{Z}$ that is an isomorphism outside $Z(\xi^{+})$ and the identity on $Z(\xi^{+})$ with $f(Z(\xi^{+}))_{o}=\emptyset$ . The map $\hat{\kappa}(\emptyset,S)$ is the same for $Z$ and for $\bar{Z}$ , so $\hat{\kappa}(\emptyset,S)$ is still bijective.

Consider now a general $Z$ and let $f:\hat{Z}\to Z$ be a non-singular cover. Let $\tilde{\hat{\xi}}:Z^{\prime}\to\hat{Z}$ be the morphism of curves such that $\tilde{\xi}=f\circ\tilde{\hat{\xi}}$ . The functors $f$ and $f^{\#}$ are inverse bijections between $\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(S,\xi^{+}(\{1,\ldots,n\})$ and $\bigoplus_{S^{\prime}}\operatorname{Hom}\nolimits_{{\mathcal{S}}(\hat{Z})}(S^{\prime},\tilde{\hat{\xi}}(\{1,\ldots,n\})$ (resp. $\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(\xi^{-}(\{-n,\ldots,-1\},S)$ and $\bigoplus_{S^{\prime}}\operatorname{Hom}\nolimits_{{\mathcal{S}}(\hat{Z})}(\tilde{\hat{\xi}}(\{-n,\ldots,-1\},S^{\prime})$ ), where $S^{\prime}$ runs over $n$ -elements subsets of $\hat{Z}$ such that $f(S^{\prime})=S$ . Furthermore, $\hat{\kappa}(\emptyset,S)$ is compatible with these bijections (see the proof of Lemma 8.1.14). It follows that $\hat{\kappa}(\emptyset,S)$ is bijective.

We consider now two arbitrary subsets $S$ and $T$ of $M$ . The canonical isomorphisms of Lemma 8.1.2 and of §8.1.5 fit in a commutative diagram of ${\mathbf{F}}_{2}$ -modules

where $S^{\prime}$ runs over $n$ elements subsets of $S$ . The discussion above shows that the left vertical arrow is an isomorphism, hence $\hat{\kappa}(T,S)$ is an isomorphism. ∎

Given $x_{1},x_{2}\in[-1,1]$ , the homotopy class $\tilde{\xi}([x_{1}\to x_{2}])$ is admissible if $x_{1}\leq x_{2}$ or $x_{1}\leq-\frac{1}{2}$ or $x_{2}\geq\frac{1}{2}$ . Given $x\in[-1,1]$ and $\zeta$ an admissible class of paths in $Z$ with $\zeta(1)=\tilde{\xi}(x)$ and $\tilde{\xi}([x\to 1])\cdot\zeta\neq 0$ , there is a unique $y\in[-1,1]$ such that $\zeta=\tilde{\xi}([y\to x])$ .

Let us describe now the unit of the adjunction when $n=1$ .

0PCN

Lemma 8.1.16. The unit of the adjunction $(L_{\xi^{+}}(-,-)\otimes-,R_{\xi^{-}}(-,-)\otimes-)$ is given by the morphism of bimodules whose evaluation at $(T,S)$ is

	$\displaystyle\operatorname{Hom}\nolimits_{{\mathcal{S}}(Z)}(S,T)$	$\displaystyle\to R_{\xi^{-}}(T,-)\otimes L_{\xi^{+}}(-,S)$
	$\displaystyle\gamma$	$\displaystyle\mapsto\sum_{x\in\tilde{\xi}^{-1}(S)}(\gamma_{\|S\setminus\{\tilde{\xi}(x)\}}\boxtimes\tilde{\xi}([-1\to x]))\otimes(\operatorname{id}\nolimits_{S\setminus\{\tilde{\xi}(x)\}}\boxtimes\tilde{\xi}([x\to 1])).$

0PCP

Proof. The counit of the adjunction is $\varepsilon=\kappa_{1}\circ\mathrm{mult}$ . Let $\gamma\in R_{\xi^{-}}^{\bullet}(T,S)$ . Let $\eta$ be the map defined in the lemma. We have

\eta(\operatorname{id}\nolimits_{T})=\sum_{x\in\tilde{\xi}^{-1}(T)}(\tilde{\xi}([-1\to x])\boxtimes\operatorname{id}\nolimits_{T\setminus\{\tilde{\xi}(x)\}})\otimes(\tilde{\xi}([x\to 1])\boxtimes\operatorname{id}\nolimits_{T\setminus\{\tilde{\xi}(x)\}}),

hence

(\operatorname{id}\nolimits\otimes\varepsilon)\circ(\eta\otimes\operatorname{id}\nolimits)(\gamma)=\sum_{x\in\tilde{\xi}^{-1}(T)}(\tilde{\xi}([-1\to x])\boxtimes\operatorname{id}\nolimits_{T\setminus\{\tilde{\xi}(x)\}})\cdot\kappa_{1}\bigl((\tilde{\xi}([x\to 1])\boxtimes\operatorname{id}\nolimits_{T\setminus\{\tilde{\xi}(x)\}})\cdot\gamma\bigr)

Let $x$ be the unique element of $\tilde{\xi}^{-1}(\chi(\gamma)(\xi^{-}(-1)))$ . We have $\gamma_{\xi^{-}(-1)}=\tilde{\xi}([-1\to x])$ and $\kappa_{1}\bigl((\tilde{\xi}([x\to 1])\boxtimes\operatorname{id}\nolimits_{T\setminus\{\tilde{\xi}(x)\}})\cdot\gamma\bigr)=\gamma_{|S}$ , hence