The complex of a braid word

Tensor products of two-sided finite graded projectives are again two-sided finite graded projectives. Let $M,N$ be graded $(A_m,A_m)$-bimodules finite graded projective on each side. To prove left projectivity, split $N$ as a left module by degree-zero left $A_m$-linear maps $i:N\to {⨁}_j{A}_m\{r_j\}$ and $p:{⨁}_j{A}_m\{r_j\}\to N$ with $pi=1$. The maps $1_M\otimes i$ and $1_M\otimes p$ are well-defined over $A_m$ and left $A_m$-linear for the action on $M$; they exhibit $M{\otimes }_{A_m}N$ as a left-module summand of ${⨁}_{j}M\{r_j\}$ by [L2]. Thus it is finite graded projective on the left by [L1]. To prove right projectivity, split $M$ as a right module and apply $-{\otimes }_{A_m}N$; the resulting right-linear maps exhibit the tensor product as a summand of finitely many shifts of the right-projective module $N$. No splitting is assumed bimodule-linear, and each is tensored on its valid balanced side.

The totalization of two complexes in $\mathcal{P}$ lies in $\mathcal{P}$, and is bounded. Let $R,S\in \mathcal{P}$. Every term of $R{\otimes }_{A_m}S$ is a finite direct sum of modules $R^p{\otimes }_{A_m}{S}^q$ with $p+q=n$; each summand is two-sided finite graded projective by step 1.1, and a finite direct sum of such is again such by [L1]. By [L3] the totalization is a complex of graded bimodules, and it is bounded because only finitely many pairs $(p,q)$ with $p+q=n$ occur and $R,S$ each have finitely many nonzero terms.

The claim for $k\le 1$, and the induction step. For $k=0$ the complex $R_1=A_m$ is a single copy of the diagonal bimodule in degree $0$, which is finite graded projective on both sides, so $R_1\in \mathcal{P}$. For $k=1$ the factors are $R_i$ or $R_i^{-1}$, which lie in $\mathcal{P}$ by [L4]; and in general, if $R_{{\tau }_1}\otimes \cdots \otimes R_{{\tau }_{\mathrm{\ell }}}\in \mathcal{P}$ then tensoring with the next factor, which lies in $\mathcal{P}$ by [L4], stays in $\mathcal{P}$ by step 2.1; hence by induction on $k$ the complex $R_{\sigma }$ lies in $\mathcal{P}$ for every word. Associativity of the iterated totalization up to canonical isomorphism, which is what makes the notation $R_{{\tau }_1}\otimes \cdots \otimes R_{{\tau }_k}$ unambiguous, is part of [L3].

The functor properties. By step 3.1 the complex $R_{\sigma }$ satisfies the hypothesis of [L5], so $R_{\sigma }{\otimes }_{A_m}-$ is a well-defined additive endofunctor of $C_m$, exact for the triangulations and carrying distinguished triangles to distinguished triangles, and it agrees with the derived tensor product through the identity replacements.

Conclusion and scope. The complex $R_{\sigma }$ attached to a word $\sigma$ is a bounded complex of graded $(A_m,A_m)$-bimodules with two-sided finite graded projective terms by step 3.1, and its action on $C_m$ is an exact triangulated endofunctor agreeing with derived tensor by step 4.1. The construction depends on the chosen word: nothing here compares $R_{\sigma }$ for different words representing the same braid, and no choice principle is used, the tensor products and shifts being explicit and finite.