Subsection 4.2. (0P2N) · ScalingStacks

4.2. Hochschild homology

4.2.1. Main Theorem

We put $\operatorname{HH}\nolimits_{i}=\operatorname{HH}\nolimits_{i}^{(W,S)}=\operatorname{Tor}\nolimits_{i}^{P^{{\mathrm{en}}}}(P,-):P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits\to P\operatorname{\!-modgr}\nolimits$ . This gives rise to functors $\operatorname{HH}\nolimits_{i}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)$ and to a functor $\operatorname{HH}\nolimits_{*}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)^{{\mathbf{Z}}}$ .

Given $I\subset S$ , we have a functor $\rho_{I}^{*}:D^{b}(P_{I}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}\to D^{b}(P\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ . This defines a functor from ${\mathcal{C}}{ox}$ to graded triangulated categories $(W,S)\mapsto D^{b}(P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ . Our grading here is the one coming from $P_{S}\operatorname{\!-modgr}\nolimits$ .

The following theorem is a consequence of Theorem 5.1 below.

0P20

Theorem 4.5. The functors $\operatorname{HH}\nolimits_{*}^{S}$ define a Markov $2$ -trace on finite Coxeter groups with value in ${\mathcal{C}}_{S}=D^{b}(P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$ and with $T_{S,s,+}=[-1]\{1\}$ and $T_{S,s,-}=\operatorname{Id}\nolimits$ .

Passing to homology, we obtain the following result.

0P21

Corollary 4.6. The functors $H^{*}\operatorname{HH}\nolimits_{*}^{S}$ define a Markov $2$ -trace on finite Coxeter groups with value in ${\mathcal{C}}_{S}=\bigl((P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}\bigr)^{{\mathbf{Z}}}$ and with $T_{S,s,+}=[-1]\{1\}$ and $T_{S,s,-}=\operatorname{Id}\nolimits$ .

The construction of §4.1.3 provides a Markov trace, recovering a result of Lasy [La1, La2]. Note that Webster and Williamson [WeWi] have shown that for finite Weyl groups, this is actually Gomi’s trace, as conjectured by J. Michel. That has been shown to hold also in type $I_{2}(n)$ by Lasy [La1].

0P22

Corollary 4.7 (Lasy). The following defines a Markov trace on finite Coxeter groups:

{\mathcal{B}}_{(W,S)}\ni b\mapsto\sum_{d,i,j}(-1)^{j}\dim H^{j}(\operatorname{HH}\nolimits_{i}^{S}(F_{b}))_{d}q^{-d}t^{-i}\in{\mathbf{Z}}[q^{\pm 1},t^{\pm 1}]

corresponding to $t_{+}=-t$ and $t_{-}=1$ .

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