The Rouquier complex of a braid word

The construction. For a signed word $\sigma ={\sigma }_{i_1}^{{ϵ}_1}\cdots {\sigma }_{i_r}^{{ϵ}_r},{ϵ}_k\in \{\pm 1\},1\le i_k\le n-1,$ put $F(\sigma ):=F_{i_1}^{{ϵ}_1}{\otimes }_R\cdots {\otimes }_R{F}_{i_r}^{{ϵ}_r},$ the iterated signed tensor totalization of Bounded graded bimodule complexes and signed tensor totalization of the Rouquier generator complexes of The positive and negative Rouquier generator complexes, bracketed left to right; the empty word ($r=0$) is sent to the unit complex $R$ concentrated in cohomological degree $0$. This is the Rouquier complex of the word. When ${ϵ}_k=+1$ the letter contributes the positive complex $F_{i_k}$ and when ${ϵ}_k=-1$ it contributes $F_{i_k}^{-1}$.

Structure. $F(\sigma )$ is a bounded cochain complex of graded $(R,R)$-bimodules with degree-zero differentials, concentrated in cohomological degrees $-r_{-},\dots ,r_{+}$, where $r_{-}$ and $r_{+}$ count the negative and positive letters: the term $F(\sigma {)}^m$ is the direct sum of the $\left(\genfrac{}{}{0px}{}{r}{m+r_{-}}\right)$ tensor products of one term of each factor, since the minimal choice has degree $-r_{-}$ and each upper-term choice adds one. It carries a cohomological degree and an internal grading, and the differential is the Koszul totalization differential. A positive letter contributes its unit term $R(1)$ in cohomological degree $1$ and the internal shift $(1)$ to that term; a negative letter contributes its unit term $R(-1)$ in cohomological degree $-1$ and the internal shift $(-1)$. Every term is a finite direct sum of finite tensor products of copies of $R(\pm 1)$ and $B_i$, hence is finite free on both sides (Soergel generators and Bott–Samelson products are finite free on both sides), so each term is finite graded projective as a left module and as a right module and $F(\sigma )$ defines a derived tensor functor by A bounded two-sided projective bimodule complex defines exact derived tensor functors.

Status of the notation. The notation records the chosen word $(\sigma )$ together with the chosen bracket; it asserts nothing about independence of the word, of the sign normalization or of the bracketing. That the homotopy class of $F(\sigma )$ depends only on the braid represented by $\sigma$ is the content of The Rouquier complex is well defined up to canonical homotopy equivalence, whose proof uses the braid relations of the generator complexes and could not be stated before those relations were proved.

Convention for comparison. The library's positive generator $F_i$ is Rouquier's $F_{s_i}=[A{\otimes }_{A^{s_i}}A\to A]$ with the library external shift $(1)$ on both terms, after giving the polynomial variables degree $2$: $F_i=[(R{\otimes }_{R^{s_i}}R)(1)⟶R(1)]\mathrm{.}$ Rouquier's negative complex of §3.2.4 becomes $F_i^{-1}$ after doubling internal degrees and shifting the whole complex by $(-1)$, up to the unit scalar coming from the root normalization. As recorded in The positive and negative Rouquier generator complexes, the complexes of Rouquier, Gorsky–Kivinen–Simental, Khovanov and Elias–Krasner differ from $F_i$ and $F_i^{-1}$ by homological or internal shifts, so any comparison with those sources is read through the dictionary recorded there. The empty word corresponds to the trivial braid and to the identity functor.