ScalingStacks

Proposition 18 We have a direct sum decomposition

$$\overline{C}(D_{1})=X_{1}\oplus X_{2}\oplus X_{3}$$

(112)

in the category ${{\mathrm{Kom}}}({R{\mbox{-mod}_{0}}})$ of complexes of graded $R$-modules.

The complexes $X_{2}$ and $X_{3}$ are acyclic.

The complex $X_{1}$ is isomorphic to the complex $\overline{C}(D)[-1]\{-1\}.$