Special Bott–Samelson Hom formula before reflection localization

Reduction step: by [F1] the two hom spaces $\mathrm{Hom}(B_s{\otimes }_{R}M,N)$ and $\mathrm{Hom}(M,B_s{\otimes }_{R}N)$ are isomorphic as graded $R$-modules, hence have the same graded rank; by [F2] the right hand sides of the claimed formula for the two pairs differ by the factor $H_s$ on the two sides and agree by the self-adjointness of [F3], so the formula holds for the pair $(B_s{\otimes }_{R}M,N)$ if and only if it holds for $(M,B_s{\otimes }_{R}N)$.

Shift step: the same argument with $M$ replaced by $M\{k\}$ uses $\mathrm{Hom}(M\{k\},N)\cong \mathrm{Hom}(M,N\{-k\})$ and the shift rules of [F2] to show that the formula for $(M\{k\},N)$ is equivalent to the formula for $(M,N\{-k\})$.

Base case for a Bott–Samelson target: if $M\in F_{\mathrm{\Delta }}$, the submodule ${\mathrm{\Gamma }}_{\mathrm{\ell }\ge 1}M$ is supported off $Gr(e)$, so [F6] shows every map $M\to R$ kills it. The remaining quotient is the length-zero layer ${⨁}_d{\mathrm{\Delta }}_e(d{)}^{(M:{\mathrm{\Delta }}_e(d))}$. Hence $\mathrm{Hom}(M,R)$ is its Hom module into $R$, free of rank ${\sum }_d(M:{\mathrm{\Delta }}_e(d))v^d$ by [F4]. This argument proves the required exactness directly; it does not apply Hom to an arbitrary nonsplit flag and assume surjectivity.

Induction on the word: write a nonempty Bott–Samelson target as $N=B_s{\otimes }_R{N}^{\mathrm{\prime }}$ by peeling its leftmost letter. The first biadjunction of [F1] gives $\mathrm{Hom}(M,B_s{\otimes }_R{N}^{\mathrm{\prime }})\cong \mathrm{Hom}(B_s{\otimes }_{R}M,N^{\mathrm{\prime }})$. By [F5], $B_s{\otimes }_{R}M\in F_{\mathrm{\Delta }}$, so the induction hypothesis computes the rank of the latter Hom module as $⟨h_{\mathrm{\Delta }}(B_s{\otimes }_{R}M),h_{\mathrm{\nabla }}(N^{\mathrm{\prime }})⟩$. The left recursion of [F2] and self-adjointness in [F3] turn this into $⟨H_s{h}_{\mathrm{\Delta }}(M),h_{\mathrm{\nabla }}(N^{\mathrm{\prime }})⟩=⟨h_{\mathrm{\Delta }}(M),H_s{h}_{\mathrm{\nabla }}(N^{\mathrm{\prime }})⟩=⟨h_{\mathrm{\Delta }}(M),h_{\mathrm{\nabla }}(B_s{\otimes }_R{N}^{\mathrm{\prime }})⟩$, the claimed rank for $(M,N)$.

Base case and induction for a Bott–Samelson source: if $N\in F_{\mathrm{\nabla }}$, evaluation at $1$ identifies $\mathrm{Hom}(R,N)$ with ${\mathrm{\Gamma }}_{e}N$, since a bimodule image of $1$ is annihilated by the identity graph ideal. For a flagged module no thickened graph occurs: its localization on any single graph is a direct sum of rank-one graph modules, and right torsion-freeness makes the graph relations hold already before localization. Thus ${\mathrm{\Gamma }}_{e}N$ is exactly its length-zero $\mathrm{\nabla }$ layer ${⨁}_e{\mathrm{\nabla }}_e(e{)}^{(N:{\mathrm{\nabla }}_e(e))}$, giving free rank ${\sum }_e(N:{\mathrm{\nabla }}_e(e))v^{-e}$. For a nonempty Bott–Samelson source $B_s{M}^{\mathrm{\prime }}$, biadjunction moves the first letter to the target, producing $\mathrm{Hom}(M^{\mathrm{\prime }},B_{s}N)$. The separately proved $\mathrm{\nabla }$ closure and recursion in [F2],[F5] and self-adjointness [F3] apply, and induction on the source word proves the dual formula and freeness.

Conclusion: finite direct sums of shifts of words are handled by additivity of Hom and of the canonical support layers, with the shift rule of step 1.2. Thus for every pair $(M,N)$ covered by the statement the graded rank of ${\mathrm{Hom}}_{R\text{-}R}(M,N)$ is the displayed multiplicity sum, and by the first-case induction and the dual induction the hom space is a free graded $R$-module; the shift and simple-reflection moves of steps 1.1 to 1.5 generate every Bott–Samelson word, so no further hypothesis on $N$ is used and the reflection-localization statement for arbitrary direct summands is not invoked.