The first-layer relations of a closed MOY resolution form a regular sequence

Set $A_0:=\mathbb{Q}[x_{0,1},\dots ,x_{0,m}]$, so that $\tilde{R}=A_0[x_{1,1},\dots ,x_{1,m}][x_{2,1},\dots ,x_{2,m}]\cdots [x_{r,1},\dots ,x_{r,m}]$, and let $Q_i$ be the quotient of the truncated polynomial ring $A_0[x_{k,j}:1\le k\le i]$ by the elements of the first $i$ layers. At every induction stage the unused later variables are polynomial extensions of this ring. We prove by induction on $i$ that the concatenation of the first $i$ layers is a regular sequence and that $Q_i$ is a free $\mathbb{Q}$-module, nonzero.

Assume $Q_{i-1}$ computed and nonzero, and consider layer $i$ in the ring $Q_{i-1}[x_{i,1},\dots ,x_{i,m}]$. Write $c:=x_{i-1,s}$, $d:=x_{i-1,s+1}$. The first relation ${\beta }_i=x_{i,s}+x_{i,s+1}-c-d$ is monic of degree one in $x_{i,s}$ with coefficient $1$, so it is a nonzerodivisor and its quotient is free over $Q_{i-1}[x_{i,s+1},x_{i,j}(j\mathrm{\ne }s,s+1)]$ with basis $\{1\}$. In that quotient, substituting $x_{i,s}=c+d-x_{i,s+1}$ turns the second relation into $-(x_{i,s+1}^2-(c+d)x_{i,s+1}+cd)$, which is monic of degree two in $x_{i,s+1}$ with leading coefficient $-1$; it is a nonzerodivisor. Each remaining difference $x_{i,j}-x_{i-1,j}$ is monic of degree one in the corresponding new variable $x_{i,j}$ and so is a nonzerodivisor on the successive quotients. Thus the $m$ elements of layer $i$ form a regular sequence on $Q_{i-1}[x_{i,1},\dots ,x_{i,m}]$ with nonzero quotient free over $Q_{i-1}$. Concatenating with the induction hypothesis, the first $i$ layers form a regular sequence on $\tilde{R}$, and $Q_i$ is nonzero and free over $\mathbb{Q}$.

Identify $Q_r$ with $B^{\mathrm{\prime }}(D)$. At layer $i$, the quotient is the base change along $\mathbb{Q}[c,d]\to Q_{i-1}$ of $\mathbb{Q}[c,d][y]\mathrm{/}(y^2-(c+d)y+cd)$, with $y=x_{i,s+1}$. By [L4] it is the corresponding base change of $A{\otimes }_{B}A$; explicitly $x_{i,s}\mapsto 1\otimes c$ and $x_{i,s+1}\mapsto 1\otimes d$, while the previous-layer variables act on the left. Both maps are inverse because $x_{i,s}=c+d-y$ and both modules have basis $1,y$; the differences at the other positions identify $x_{i,j}$ with the corresponding variable of the previous layer without changing the ring. Iterating over the layers, the surviving ring is generated by all layer variables subject to the displayed invariant-balancing relations and is the balanced tensor product over the shared strand variables of one two-variable balanced tensor $A{\otimes }_{B}A$ per wide edge; each such tensor is the two-variable case of the unreduced simple-reflection bimodule $B_s^{\mathrm{\prime }}$ of Unreduced type-A Soergel bimodules and the trivial polynomial factor, and the iteration over layers is the balanced tensor product over the shared variables defining $B^{\mathrm{\prime }}(D)={⨂}_{j=1}^r{B}_{s_j}^{\mathrm{\prime }}$.

Conclude the resolution statement. By steps 1.1 and 2.1 the first $rm$ elements are $\tilde{R}$-regular and their quotient is $B^{\mathrm{\prime }}(D)$, which is nonzero; since $K(\text{first }rm;\tilde{R})$ is a finite free complex by its definition and a regular sequence has acyclic positive Koszul homology, [L5] exhibits it as a finite free resolution of the quotient $B^{\mathrm{\prime }}(D)$. The closure differences are not included in the sequence and nothing is claimed about their regularity.