ScalingStacks

Corollary 3.14. The filtration-preserving component of the chain maps

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

agree. More precisely, we have

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

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