Subsection 4.2.2. (0P2Q) · ScalingStacks

4.2.2. Shift adjustment

By shifting suitably the invariants, we can get rid of the automorphisms $T_{S,s,\pm}$ , but we lose functoriality (it would be interesting to see if functoriality with respect to an appropriate notion of cobordisms can be implemented). In order to do this, we need to use $\frac{1}{2}{\mathbf{Z}}$ -complexes.

Given ${\mathcal{A}}$ an additive category, the category of $\frac{1}{2}$ -complexes in ${\mathcal{A}}$ has objects $(C^{i},d^{i})_{i\in\frac{1}{2}{\mathbf{Z}}}$ where the differential has degree $1$ , and morphisms are $\frac{1}{2}{\mathbf{Z}}$ -graded maps commuting with the differential. Its homotopy category is denoted by $\operatorname{Ho}\nolimits_{\frac{1}{2}}({\mathcal{A}})$ and, when ${\mathcal{A}}$ is an abelian category, its derived category by $D_{\frac{1}{2}}({\mathcal{A}})$ .

0P23

Corollary 4.8. Given $(W,S)$ a finite Coxeter group and $b\in B_{S}$ , let

N_{S}(F_{b})=\operatorname{HH}\nolimits_{*-\frac{|S|+l(b)}{2}}^{S}(F_{b})\left[\frac{|S|+l(b)}{2}\right]\in D^{b}_{\frac{1}{2}}(P_{S}\operatorname{\!-modgr}\nolimits)^{\frac{1}{2}{\mathbf{Z}}}.

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We have $N_{S}(F_{b}F_{b^{\prime}})\simeq N_{S}(F_{b^{\prime}}F_{b})$ for all $b,b^{\prime}\in B_{S}$ .
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Given $s\in S$ and $b\in B_{S\setminus s}$ , we have $N_{S}(\gamma_{S\setminus s}(F_{b})F_{s}^{\pm 1})\simeq\rho_{S\setminus s}^{*}N_{S\setminus s}(F_{b})$ .

Raphaël Rouquier

Original source: arXiv:1203.5065v1