The positive and negative Rouquier generator complexes

The setting. Keep $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$, the place-permutation action of $S_n$, the balanced roots ${\alpha }_i={\epsilon }_i(x_i-x_{i+1})$ with ${\epsilon }_i=(-1{)}^{i-1}$ and the graded $(R,R)$-bimodules $B_i=R{\otimes }_{R^{s_i}}R(1)$ of The Soergel bimodule $B_i$ of a simple reflection; recall that $1\otimes 1\in B_i$ has degree $-1$ and $1\otimes {\alpha }_i$ has degree $1$, that $B_i$ is generated as an $(R,R)$-bimodule by $1\otimes 1$, and that $R=R^{s_i}\oplus {\alpha }_i{R}^{s_i}$ with $s_i({\alpha }_i)=-{\alpha }_i$ (The standard type-A reflection realization and its polynomial ring, The Soergel bimodule $B_i$ of a simple reflection).

The positive generator complex. For $1\le i\le n-1$ let $F_i$ be the bounded cochain complex of graded $(R,R)$-bimodules $F_i:=[B_i\stackrel{{\epsilon }_i}{\to }R(1)],$ with $B_i$ in cohomological degree $0$, $R(1)$ in cohomological degree $1$ and all other terms zero, where ${\epsilon }_i:B_i\to R(1)$ is the multiplication map ${\epsilon }_i(r\otimes r^{\mathrm{\prime }})=r{r}^{\mathrm{\prime }}$, i.e. the degree-zero bimodule map determined by ${\epsilon }_i(1\otimes 1)=1$ and ${\epsilon }_i(1\otimes {\alpha }_i)={\alpha }_i$.

The negative generator complex. Let $F_i^{-1}$ be the bounded cochain complex of graded $(R,R)$-bimodules $F_i^{-1}:=[R(-1)\stackrel{{\eta }_i}{\to }{B}_i],$ with $R(-1)$ in cohomological degree $-1$, $B_i$ in cohomological degree $0$ and all other terms zero, where ${\eta }_i:R(-1)\to B_i$ is the bimodule map ${\eta }_i(1)={\alpha }_i\otimes 1+1\otimes {\alpha }_i$.

Well-definedness of the differentials. The map ${\epsilon }_i$ descends from the multiplication $R{\otimes }_{\mathbb{Q}}R\to R$ because ${\epsilon }_i(ra\otimes r^{\mathrm{\prime }})=ra{r}^{\mathrm{\prime }}={\epsilon }_i(r\otimes a{r}^{\mathrm{\prime }})$ for $a\in R^{s_i}$ and $R$ is commutative; it is left and right $R$-linear and homogeneous of degree zero, since $1\otimes 1\mapsto 1$ matches the degrees $-1$ of $B_i$ and of the generator of $R(1)$, and $1\otimes {\alpha }_i\mapsto {\alpha }_i$ matches the degree $1=\mathrm{deg}({\alpha }_i)-1$ on both sides. For ${\eta }_i$, the bimodule map $R\to B_i$ sending $1$ to the class of $x_i-x_{i+1}^{\mathrm{\prime }}$ is well defined: the products $(x_i-x_{i+1}^{\mathrm{\prime }})(x_i-x_i^{\mathrm{\prime }})$ and $(x_i-x_{i+1}^{\mathrm{\prime }})(x_{i+1}-x_{i+1}^{\mathrm{\prime }})$ vanish in $B_i$ by the explicit computation of GKS Lemma 3.8, while for $a\mathrm{\notin }\{i,i+1\}$ the factor $x_a-x_a^{\mathrm{\prime }}$ is already a defining relation of $B_i$, so the element annihilates the kernel ideal of the multiplication $R{\otimes }_{\mathbb{Q}}R\to R$, and the assignment $1\mapsto x_i-x_{i+1}^{\mathrm{\prime }}$ extends to a well-defined $R$-bimodule map. The value of that map at $1$ times the unit $2{\epsilon }_i$ is ${\eta }_i(1)$, because $1\otimes (x_i-x_{i+1})+(x_i-x_{i+1})\otimes 1=2(x_i-x_{i+1}^{\mathrm{\prime }})$ in $B_i$ (GKS Lemma 3.8, using $x_i+x_{i+1}=x_i^{\mathrm{\prime }}+x_{i+1}^{\mathrm{\prime }}$) and ${\alpha }_i={\epsilon }_i(x_i-x_{i+1})$; a unit multiple of a well-defined bimodule map is a well-defined bimodule map, so ${\eta }_i$ is well defined. Its value ${\eta }_i(1)$ is homogeneous of degree $1$, matching the degree of $1\in R(-1)$, so ${\eta }_i$ is a degree-zero bimodule map. Both complexes are concentrated in two adjacent cohomological degrees, so $d^2=0$ holds trivially and each of $F_i,F_i^{-1}$ is a bounded cochain complex of graded $(R,R)$-bimodules in the sense of Bounded graded bimodule complexes and signed tensor totalization.

Size and consequences. The bimodules $R$, $R(1)$ and $R(-1)$ are free of rank one on both sides and $B_i$ is free of rank two on both sides (Soergel generators and Bott–Samelson products are finite free on both sides, the bases being $\{1\otimes 1,1\otimes {\alpha }_i\}$ on the left and $\{1\otimes 1,{\alpha }_i\otimes 1\}$ on the right). Every term of $F_i$ and of $F_i^{-1}$ is therefore finite free, hence finite graded projective, as a left $R$-module and as an underlying right $R$-module. Applying A bounded two-sided projective bimodule complex defines exact derived tensor functors with the commutative ring $\mathbb{Q}$ and $A=B=R$: the signed totalization with $F_i$ is an exact triangulated functor on bounded complexes of finite graded projective $R$-modules, preserves quasi-isomorphisms between bounded complexes, and descends to the derived tensor functor $F_i{\otimes }_R^{\mathbf{L}}-$ on $D^b(R\text{-grmod})$; the same statements hold with $F_i^{-1}$ in place of $F_i$. The right-tensor construction $-{\otimes }_R{F}_i$ is defined separately by signed totalization, and its derived version uses the same two-sided freeness. Commutativity of $R$ does not assert a symmetry between tensor products of arbitrary $(R,R)$-bimodules.

Recorded convention. The complex $F_i$ is the library normalization of Rouquier's positive complex $F_s=[A{\otimes }_{A^s}A\to A]$, in which $A$ sits in degree $1$ and the differential is multiplication: in the library external shift $(r)$ with $M(r{)}_d=M_{d+r}$ one has $F_i=T_i(1)$ and $F_i^{-1}\cong T_i^{-1}(-1)$ for the GKS complexes $T_i=[B_i(-1)\to R]$, $T_i^{-1}=[R\to B_i(1)]$ of formula (3.3). The GKS negative differential sends $1$ to $x_i-x_{i+1}^{\mathrm{\prime }}$, whereas ours is $2{\epsilon }_i$ times that map. The chain isomorphism $F_i^{-1}\to T_i^{-1}(-1)$ is multiplication by $2{\epsilon }_i$ in degree $-1$ and the identity in degree $0$. The internal shifts cancel in inverse pairs and agree on the two sides of each positive braid relation. The sign ${\epsilon }_i$ in ${\alpha }_i={\epsilon }_i(x_i-x_{i+1})$, and with it the sign of ${\eta }_i$, is a unit of $\mathbb{Q}$; the homotopy classes of the complexes do not depend on the choice of balanced root.