Subsection 8.2.6. (0PIC) · ScalingStacks

8.2.6. Large enough $M$

We assume in §8.2.6 that $(\xi_{1}^{+})^{-1}(M)$ has no maximum and $(\xi_{2}^{-})^{-1}(M)$ has no minimum. Fix an increasing sequence $(m^{+}_{0},m^{+}_{1},\ldots)$ of points of $(\xi_{1}^{+})^{-1}(M)$ and a decreasing sequence $(m^{-}_{0},m^{-}_{1},\ldots)$ of points of $(\xi_{2}^{-})^{-1}(M)$ such that $\lim_{i}m^{+}_{i}>t$ for all $t\in(\xi_{1}^{+})^{-1}(M)$ and $\lim_{i}m^{-}_{i}<t$ for all $t\in(\xi_{2}^{-})^{-1}(M)$ .

0PDN

Lemma 8.2.17. We have a canonical isomorphism $LR_{\xi_{2}^{-}}^{\bullet}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}G_{1}$ .

0PDP

Proof. Using (8.1.1) and (8.1.5), we have isomorphisms

L_{\xi_{1}^{+}}^{\bullet}(T,-)\wedge R_{\xi_{2}^{-}}^{\bullet}(-,S)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\\ \operatorname{colim}\nolimits_{r,s\to\infty}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(-,T\sqcup\{\xi_{1}^{+}(m^{+}_{r})\})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(m^{-}_{s})\},-)\\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{colim}\nolimits_{r,s\to\infty}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(m^{-}_{s})\},T\sqcup\{\xi_{1}^{+}(m^{+}_{r})\})\\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup\{\xi_{2}^{-}(-1)\},T\sqcup\{\xi_{1}^{+}(1)\}).

and the lemma follows. ∎

Let us define $\lambda:R_{\xi_{2}^{-}}^{\bullet}L\to LR_{\xi_{2}^{-}}^{\bullet}$ as the composition of the injective map $\mu_{1}:R_{\xi_{2}^{-}}^{\bullet}L\to G_{1}$ (cf Lemma 8.2.5) with the inverse of the isomorphism of the lemma above.

Under the assumptions above, we have a simpler version of Theorem 8.2.1.

0PDQ

Theorem 8.2.18. The functor $\tilde{\Xi}$ factors through $\Delta^{\prime}_{\lambda}{\mathcal{S}}^{\bullet}_{M}(Z)$ and induces an isomorphism of differential pointed categories $\Delta^{\prime}_{\lambda}{\mathcal{S}}^{\bullet}_{M}(Z)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}^{\bullet}_{M}(Z_{\xi})$ .

0PDR

Proof. Every element of $R_{\xi_{2}}^{-}(S,T)$ is of the form $(\operatorname{id}\nolimits_{T}\boxtimes\zeta)\cdot(\alpha\boxtimes\operatorname{id}\nolimits_{-1})$ for some $\zeta$ admissible class of paths starting at $-1$ and $\alpha$ a braid starting at $S$ .

Every element of $L_{\xi_{1}}^{+}(S,T)$ is of the form $([m_{i}^{+}\to 1]\boxtimes\operatorname{id}\nolimits_{T})\cdot\alpha$ for some braid $\alpha$ starting at $S$ .

It follows that every element of $(R_{\xi_{2}}^{-}L_{\xi_{1}}^{+})^{n}$ is of the form

(\operatorname{id}\nolimits\boxtimes\zeta_{1})\wedge([m_{i}^{+}\to 1]\boxtimes\operatorname{id}\nolimits)\wedge\cdots\wedge(\operatorname{id}\nolimits\boxtimes\zeta_{n-1})\wedge([m_{i+n-2}^{+}\to 1]\boxtimes\operatorname{id}\nolimits)\wedge(\operatorname{id}\nolimits\boxtimes\zeta_{n})\wedge\alpha

for some $i\geq 0$ and $\zeta_{r}$ an admissible class of paths starting at $-1$ for $1\leq r\leq n$ . The image by $\mu_{n}$ of such an element is

\bigl(([m_{i+r-1}^{+}\to r])_{1\leq r\leq n-1}\boxtimes\operatorname{id}\nolimits\bigr)\circ\alpha)\boxtimes\zeta_{1}\boxtimes(\zeta_{2}\circ[-2\to-1])\boxtimes\cdots\boxtimes(\zeta_{n}\circ[-n\to-1]).

It follows that $\mu_{n}$ is injective, hence it induces an isomorphism $(R_{\xi_{2}}^{-}L_{\xi_{1}}^{+})^{n}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{n}$ .

Let $L$ be the image of $\lambda\circ(T_{1}\otimes 1-1\otimes T_{1})$ . We have $\nu_{2}=\mu_{2}\circ\lambda$ . It follows that $\mu_{2}(L)=(T_{1}\otimes 1-1\otimes T_{1})(D_{2})$ , since $D_{2}$ is the image of $\nu_{2}$ (Lemma 8.2.5). The theorem follows now from Corollary 8.2.16 and Theorem 8.2.1. ∎

0PDS

Remark 8.2.19. Consider $Z$ the singular curve quotient of oriented ${\mathbf{R}}$ by the identification of two points. Take $M$ to be the single exceptional point of $Z$ . The construction above applied to ${\mathcal{S}}_{M}^{\bullet}(Z)$ gives a category where going twice around the circle, avoiding the loop, is non-zero (cf picture below), while it is not represented by a smooth path in $Z_{\xi}$ . Theorem 8.2.18 does not hold because $M$ is too small.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2

8.2.6. Large enough MM

8.2.6. Large enough $M$