Rouquier generator complexes have canonical derived graph models

Given: A simple reflection $s_i$, the bimodule $B_i$ with generators $u=1\otimes 1$ (degree $-1$) and $w_0=1\otimes {\delta }_i$, ${\delta }_i={\alpha }_i\mathrm{/}2$, the element ${\rho }_{-}={\delta }_{i}u-w_0$, and the complexes $F_i,F_i^{-1}$ of The positive and negative Rouquier generator complexes.

The two exact sequences. The multiplication ${\epsilon }_i:B_i\to R(1)$ is a surjective degree-zero bimodule map with ${\epsilon }_i(u)=1$ and ${\epsilon }_i(w_0)={\delta }_i$, and $\ker {\epsilon }_i=R{\rho }_{-}$ with $R{\rho }_{-}\cong R_{s_i}(-1)$ a graded sub-bimodule generated in degree $1$; the map ${\eta }_i:R(-1)\to B_i$ is injective and the quotient $B_i\mathrm{/}{\eta }_i(R(-1))=B_i\mathrm{/}R{\rho }_{+}$ is generated by the image of $u$ with right action $u\cdot g\equiv s_i(g)u$, so that it is isomorphic to $R_{s_i}(1)$; here ${\rho }_{+}={\delta }_{i}u+w_0$ satisfies ${\rho }_{+}\cdot g=g{\rho }_{+}$ and ${\rho }_{-}\cdot g=s_i(g){\rho }_{-}$ (Standard graph bimodules, support filtrations and characters, The positive and negative Rouquier generator complexes).

Two canonical chain maps. The assignment $1\mapsto 1\otimes {\alpha }_i-{\alpha }_i\otimes 1=-2{\rho }_{-}$ defines a degree-zero bimodule map $f_i:R_{s_i}(-1)\to F_i$, concentrated in cohomological degree $0$; it is a chain map because ${\epsilon }_i({\rho }_{-})=0$. The assignment $1\otimes 1\mapsto 1$ defines the twisted multiplication $\psi :B_i\to R_{s_i}(1)$, $\psi (r\otimes r^{\mathrm{\prime }})=r{s}_i(r^{\mathrm{\prime }})$, a degree-zero bimodule map with $\psi ({\eta }_i(1))=0$; on the quotient $B_i\mathrm{/}{\eta }_i(R(-1))$ it induces the identification with $R_{s_i}(1)$ and defines a chain map $g_i:F_i^{-1}\to R_{s_i}(1)$, concentrated in cohomological degree $0$. (Standard graph bimodules, support filtrations and characters, Canonical comparisons between standard graph tensor products)

Cohomology of the generator complexes. $H^0(F_i)=\ker {\epsilon }_i=R{\rho }_{-}\cong R_{s_i}(-1)$, $H^1(F_i)=\mathrm{coker}{\epsilon }_i=0$, and $H^{-1}(F_i^{-1})=\ker {\eta }_i=0$, $H^0(F_i^{-1})=B_i\mathrm{/}{\eta }_i(R(-1))\cong R_{s_i}(1)$; both complexes are otherwise concentrated in the displayed degrees. [F1]

Localization. The localization functor $K^b(R^e\text{-grmod})\to D^b(R^e\text{-grmod})$ sends quasi-isomorphisms to isomorphisms, and tensor totalization with a bounded complex of bimodules flat on the tensoring side preserves quasi-isomorphisms. Here all generator complexes and graph models are flat on both sides: the former have finite-free terms and the latter are twisted rank-one regular modules. Consequently the tensor comparisons used below are compatible with localization (The localization functor sends quasi isomorphisms to isomorphisms, Derived category of an abelian category, Bounded above flat tensor complexes preserve quasi isomorphisms).

Multiplication of graph bimodules. The balanced assignment $R_x{\otimes }_R{R}_y\to R_{xy}$, $a\otimes b\mapsto ax(b)$, is a degree-zero isomorphism of graded bimodules with inverse $1\mapsto 1\otimes 1$; shifts satisfy $M(a){\otimes }_{R}N(b)\cong (M{\otimes }_{R}N)(a+b)$ (Canonical comparisons between standard graph tensor products, Standard graph bimodules, support filtrations and characters).