ScalingStacks

Note that the $\widetilde{\mathcal{L}}$-resolution $D(\widetilde{\mathcal{L}})$ of diagram $D$ and the $\mathcal{L}$-resolution $D^{!}(\mathcal{L})$ of $D^{!}$ are isomorphic. Let $k$ be the number of circles in $D(\widetilde{\mathcal{L}}).$ Then

$${\cal F}(D(\widetilde{\mathcal{L}}))={\cal F}(D^{!}(\mathcal{L}))={\cal A}^{\otimes k}$$

(162)

and, via $\mu,$ we can identify

$${\cal F}(D(\widetilde{\mathcal{L}}))=({\cal A}^{\ast})^{\otimes k}=({\cal F}(D^{!}(\mathcal{L})))^{\ast}$$

(163)

Since $\mu$ maps $m,\Delta$ to $m^{\ast},\Delta^{\ast},$ we see that after suitable shifts (recall that ${\cal V}_{D}(\mathcal{L})$ is equal to ${\cal F}(D(\mathcal{L}))$ shifted up by $|\mathcal{L}|$) the identification (163) extends to an isomorphism of $\mathcal{I}$-cubes ${\cal V}_{D^{!}}\{-n\}$ and $({\cal V}_{D})^{\ast}.$