ScalingStacks

Proposition 3.13. The filtration-preserving components of the chain maps

$$R3_{+}\colon\left\llbracket L^{i}_{+}\right\rrbracket\to\left\llbracket L^{i+1}_{+}\right\rrbracket\quad\text{and}\quad R3_{-}\colon\left\llbracket L^{i}_{-}\right\rrbracket\to\left\llbracket L^{i+1}_{-}\right\rrbracket$$

agree if $1\leq i<x-1$. More precisely, we have

$$R3_{+}(W^{i}_{+},W^{i+1}_{+})=R3_{-}(W^{i}_{-},W^{i+1}_{-})$$

for pairs of corresponding webs $W^{i}_{\pm}$ in $\left\llbracket L^{i}_{\pm}\right\rrbracket$ and $W^{i+1}_{\pm}$ in $\left\llbracket L^{i+1}_{\pm}\right\rrbracket$ with $\mathrm{gr}_{\mathrm{ext}}(W^{i}_{\pm})=\mathrm{gr}_{\mathrm{ext}}(W^{i+1}_{\pm})$.