ScalingStacks

Lemma 3.10. The chain maps $R2_{\pm}\colon\left\llbracket L^{0}_{\pm}\right\rrbracket\to\left\llbracket L^{1}_{\pm}\right\rrbracket$ and $R2^{-1}_{\pm}\colon\left\llbracket L^{x-1}_{\pm}\right\rrbracket\to\left\llbracket L^{x}_{\pm}\right\rrbracket$ preserve the internal and external homological gradings individually and their only non-zero components involve palindromic resolutions.