The remaining closure Koszul complex is the diagonal Hochschild complex

Identify the surviving elements. The variables $x_{0,j}$ and $x_{r,j}$ are elements of the commutative ring $B^{\mathrm{\prime }}(D)=\tilde{R}\mathrm{/}(\text{first }rm)$; the left $R^{\mathrm{\prime }}$-action on $B^{\mathrm{\prime }}(D)$ is the algebra map $R^{\mathrm{\prime }}\to B^{\mathrm{\prime }}(D)$, $x_j\mapsto x_{0,j}$, and the right action is $x_j\mapsto x_{r,j}$, both induced by the ring structure. Hence the class of $x_{0,j}-x_{r,j}$ is the image of $u_j=x_j^L-x_j^R$ under the induced map $R^{\mathrm{\prime }e}\to B^{\mathrm{\prime }}(D)$, and its action on $B^{\mathrm{\prime }}(D)$ by multiplication is $m\mapsto x_{0,j}m-m{x}_{r,j}=u_j\cdot m$.

Reduce the full Koszul complex to the closure complex. By [L4] the Koszul complex of all $(r+1)m$ elements is the tensor product over $\tilde{R}$ of the Koszul complex of the first $rm$ elements and that of the last $m$ elements. By [L1] the first factor is a free resolution of $B^{\mathrm{\prime }}(D)$, and its augmentation has acyclic mapping cone. Tensor that cone with the bounded free complex $K(\text{last};\tilde{R})$ and filter the total complex by the latter factor's degree. Each associated graded complex is a finite direct sum of shifts of the acyclic cone. The filtration is finite, so induction through its short exact sequences makes the tensor cone acyclic; thus tensoring the augmentation preserves quasi-isomorphism; therefore the Koszul complex of all $(r+1)m$ elements is quasi-isomorphic to $B^{\mathrm{\prime }}(D){\otimes }_{\tilde{R}}K(\text{last};\tilde{R})\cong K(\text{last};B^{\mathrm{\prime }}(D))$, the Koszul complex of the classes of the closure differences over the commutative ring $B^{\mathrm{\prime }}(D)$.

Identify it with the diagonal complex. Under the identification of the tensor factors, $K(\text{last};B^{\mathrm{\prime }}(D))$ has degree-$p$ term the $p$-fold wedge over $B^{\mathrm{\prime }}(D)$ of the elements $x_{0,j}-x_{r,j}$ with the Koszul differential of [L4], and by step 1.1 these elements act by multiplication as $u_j=x_j^L-x_j^R$. By [L3] the diagonal Koszul bimodule complex tensored over $R^{\mathrm{\prime }e}$ with $B^{\mathrm{\prime }}(D)$ has degree-$p$ term ${⨁}_{\mathrm{\mid }I\mathrm{\mid }=p}{B}^{\mathrm{\prime }}(D){\theta }_I$ and differential $d(m{\theta }_I)={\sum }_r(-1{)}^{r-1}(x_{i_r}m-m{x}_{i_r}){\theta }_{I\setminus \{i_r\}}$, which in the commutative ring $B^{\mathrm{\prime }}(D)$ is multiplication by the classes $x_{0,j}-x_{r,j}$; both complexes therefore have the same graded terms and the same differential, so $K(\text{last};B^{\mathrm{\prime }}(D))$ is the diagonal Koszul bimodule complex of $B^{\mathrm{\prime }}(D)$ over $R^{\mathrm{\prime }}$.

Compute the homology. The bimodule $B^{\mathrm{\prime }}(D)$ is a $\mathbb{Q}$-algebra bimodule, hence $k$-central for $k=\mathbb{Q}$, and it is graded with $\mathrm{deg}{x}_{i,j}=2$. Applying [L3] to $M=B^{\mathrm{\prime }}(D)$ gives the natural isomorphism $H{H}_h(R^{\mathrm{\prime }},B^{\mathrm{\prime }}(D))\cong H_h(K(\text{last};B^{\mathrm{\prime }}(D)))$ and, through steps 2.1 and 2.2, the identification of $H{H}_{\bullet }(R^{\mathrm{\prime }},B^{\mathrm{\prime }}(D))$ with the homology of the Koszul complex of all $(r+1)m$ elements. This is the only place where AC is used.