Lemma 8.3.3 (0PDW) · ScalingStacks

Lemma 8.3.3. Given $\alpha\in L_{\xi_{1}^{+}}^{\bullet}(T,U)$ and $\beta\in R_{\xi_{1}^{-}}^{\bullet}(U,S)$ , we have

\rho(\alpha\otimes\beta)=\delta_{1}\bigr(\alpha_{|U\setminus\chi(\alpha)^{-1}(\xi_{1}^{+}(1))}\cdot(\beta_{\xi_{1}^{-}(-1)}\boxtimes\operatorname{id}\nolimits)\bigl)\otimes\bigl((\alpha_{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))}\boxtimes\operatorname{id}\nolimits)\cdot\beta_{|S}\bigr)

where $\delta_{1}=1$ if $\chi(\alpha\circ\beta)(\xi_{1}^{-}(-1))\neq\xi_{1}^{+}(1)$ and $(\operatorname{id}\nolimits_{\chi(\beta)(\xi_{1}^{-}(-1))}\boxtimes\alpha_{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))})\cdot(\beta_{\xi_{1}^{-}(-1)}\boxtimes\operatorname{id}\nolimits_{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))})\neq 0$ and $\delta_{1}=0$ otherwise.

0PDX

Proof. Assume first $\alpha_{|U\setminus\{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))\}}=\operatorname{id}\nolimits$ and $\beta_{|S}=\operatorname{id}\nolimits$ . We have

	$\displaystyle\rho(\alpha\otimes\beta)$	$\displaystyle=\varepsilon_{1}R_{\xi_{1}^{-}}L_{\xi_{1}^{+}}\circ L_{\xi_{1}^{+}}\tau L_{\xi_{1}^{+}}(\alpha\otimes\beta\otimes\eta_{1}(\operatorname{id}\nolimits_{S}))$
		$\displaystyle=\varepsilon_{1}R_{\xi_{1}^{-}}L_{\xi_{1}^{+}}\biggl(\sum_{x\in\tilde{\xi}_{1}^{-1}(S)}\alpha\otimes\tau\Bigl((\beta_{\xi_{1}^{-}(-1)}\boxtimes\operatorname{id}\nolimits)\otimes\bigl(\operatorname{id}\nolimits\boxtimes\tilde{\xi}_{1}([-1\to x])\bigr)\Bigr)\otimes\bigl(\tilde{\xi}_{1}([x\to 1])\boxtimes\operatorname{id}\nolimits\bigr)\biggr)$
		$\displaystyle=\varepsilon_{1}R_{\xi_{1}^{-}}L_{\xi_{1}^{+}}\biggl(\sum_{x\in I}\alpha\otimes\bigl(\operatorname{id}\nolimits\boxtimes\tilde{\xi}_{1}([-1\to x])\bigr)\otimes(\beta_{\xi_{1}^{-}(-1)}\boxtimes\operatorname{id}\nolimits)\otimes\bigl(\tilde{\xi}_{1}([x\to 1])\boxtimes\operatorname{id}\nolimits\bigr)\biggr)$
		$\displaystyle=\delta_{1}(\beta_{\xi_{1}^{-}(-1)}\boxtimes\operatorname{id}\nolimits)\otimes(\alpha_{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))}\boxtimes\operatorname{id}\nolimits)$

where $I=\{x\in\tilde{\xi}_{1}^{-1}(S)\ |\ (\tilde{\xi}_{1}([-1\to x])\boxtimes(\beta_{\xi_{1}^{-}(-1)}\circ[\xi_{1}^{-}(-2)\to\xi_{1}^{-}(-1)])\cdot\tau\neq 0\}$ .

Since $\rho$ is a morphism of $({\mathcal{S}}_{M}(Z),{\mathcal{S}}_{M}(Z))$ -bimodules, the general result follows using the decompositions $\alpha=(\alpha_{|U\setminus\{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))\}}\boxtimes\operatorname{id}\nolimits_{\xi_{1}^{+}(1)})\cdot(\operatorname{id}\nolimits\boxtimes\alpha_{\chi(\alpha)^{-1}(\xi_{1}^{+}(1))})$ and $\beta=(\operatorname{id}\nolimits\boxtimes\beta_{\xi_{1}^{-}(-1)})\cdot(\beta_{|S}\boxtimes\operatorname{id}\nolimits_{\xi_{1}^{-}(-1)})$ . ∎