ScalingStacks

Using the splitting (89) of $V_{D_{2}}$ and Lemma 1, we can decompose the $\mathcal{I}^{\prime}$-cube $V_{D_{1}}$ as a direct sum of two $\mathcal{I}^{\prime}$-cubes as follows:

$$V_{D_{1}}=V^{\prime}\oplus V^{\prime\prime}$$

(95)

where

	$\displaystyle V^{\prime}(\ast 0)$	$\displaystyle=$	$\displaystyle 0$		(96)
	$\displaystyle V^{\prime}(\ast 1)$	$\displaystyle=$	$\displaystyle\aleph(V_{D})\{-1\}\subset V_{D_{2}}\{-1\}=V_{D_{1}}(\ast 1)$		(97)
	$\displaystyle V^{\prime\prime}(\ast 0)$	$\displaystyle=$	$\displaystyle V_{D}=V_{D_{1}}(\ast 0)$		(98)
	$\displaystyle V^{\prime\prime}(\ast 1)$	$\displaystyle=$	$\displaystyle\Delta_{a}(V_{D})\{-1\}\subset V_{D_{2}}\{-1\}=V_{D_{1}}(\ast 1)$		(99)

Tensoring (95) with $E_{\mathcal{I}^{\prime}}$ we get a splitting of skew-commutative $\mathcal{I}^{\prime}$-cubes

$$V_{D_{1}}\otimes E_{\mathcal{I}^{\prime}}=(V^{\prime}\otimes E_{\mathcal{I}^{\prime}})\oplus(V^{\prime\prime}\otimes E_{\mathcal{I}^{\prime}})$$

(100)

This induces a splitting of complexes associated to these skew $\mathcal{I}^{\prime}$-cubes

$$\overline{C}(V_{D_{1}}\otimes E_{\mathcal{I}^{\prime}})=\overline{C}(V^{\prime}\otimes E_{\mathcal{I}^{\prime}})\oplus\overline{C}(V^{\prime\prime}\otimes E_{\mathcal{I}^{\prime}})$$

(101)