ScalingStacks

We decompose $\overline{C}(D_{1})$ and $\overline{C}(D_{2})$ into following direct sums:

$$\overline{C}(D_{i})=\overline{C}(D_{i}(\ast 1))[-1]\{-1\}\oplus{\mathop{\oplus}\limits_{u,v\in\{0,1\}}}\overline{C}(D_{i}(\ast uv0))[-u-v]\{-u-v\}$$

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These are direct sum decompositions of $\mathbb{Z}\oplus\mathbb{Z}$-graded $R$-modules, not complexes. The diagrams $D_{i}(\ast uv0)$ for $i=1,2$ and $u,v\in\{0,1\}$ are depicted below