Subsection 3.2. (0P2G) · ScalingStacks

Let us recall, following [Rou1], how to lift in a non-trivial way the action of $W$ on the derived category $D(P)$ to an action of $B_{S}$ on the homotopy category $\operatorname{Ho}\nolimits(P)$ .

Let $s\in S$ . We put

\theta_{s}=P\otimes_{P^{s}}P\text{ and }F_{s}=0\to\theta_{s}\langle 1\rangle\xrightarrow{m}P\langle 1\rangle\to 0.

The latter is a complex of graded $P^{{\mathrm{en}}}$ -modules, where $P\langle 1\rangle$ is in cohomological degree $1$ and $m$ denotes the multiplication map. We put

F_{s}^{-1}=0\to P\langle-1\rangle\xrightarrow{a\mapsto a\alpha_{s}\otimes 1+a\otimes\alpha_{s}}\theta_{s}\to 0.

This is a complex of graded $P^{{\mathrm{en}}}$ -modules, where $P\langle-1\rangle$ is in cohomological degree $-1$ .

Let us recall a result of [Rou1, §9]. Given $i_{1},\ldots,i_{r}$ , $j_{1},\ldots,j_{r^{\prime}}\in S$ and $\delta_{1},\ldots,\delta_{r}$ , $\varepsilon_{1},\ldots,\varepsilon_{r^{\prime}}\in\{\pm 1\}$ such that $\sigma_{i_{1}}^{\delta_{1}}\cdots\sigma_{i_{r}}^{\delta_{r}}=\sigma_{j_{1}}^{\varepsilon_{1}}\cdots\sigma_{j_{r^{\prime}}}^{\varepsilon_{r^{\prime}}}$ , there is a canonical isomorphism in $\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$

F_{i_{1}}^{\delta_{1}}\otimes_{P_{n}}\cdots\otimes_{P_{n}}F_{i_{r}}^{\delta_{r}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}F_{j_{1}}^{\varepsilon_{1}}\otimes_{P_{n}}\cdots\otimes_{P_{n}}F_{j_{r^{\prime}}}^{\varepsilon_{r^{\prime}}}

and these isomorphisms form a transitive system of isomorphisms.

Given $b\in B_{S}$ , we put

F_{b}=\lim_{\begin{subarray}{c}i_{1},\ldots,i_{r}\\ \varepsilon_{1},\ldots,\varepsilon_{r}\\ b=\sigma_{i_{1}}^{\varepsilon_{1}}\cdots\sigma_{i_{r}}^{\varepsilon_{r}}\end{subarray}}F_{i_{1}}^{\varepsilon_{1}}\otimes_{P}\cdots\otimes_{P}F_{i_{r}}^{\varepsilon_{r}}\in\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits).

The $2$ -braid group ${\mathcal{B}}_{(W,S)}$ is the full monoidal subcategory of $\operatorname{Ho}\nolimits(P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits)$ with objects the $F_{b}$ ’s, with $b\in B_{S}$ .

3.2. Lift