ScalingStacks

0P7X

Proof. Assume first I=J=KI=J=K. The lemma follows in that case from Lemmas 3.2.4 and 3.2.2.

Consider now the general case. There are increasing bijections τ:J→I\tau:J\to I and τ′:K→J\tau^{\prime}:K\to J. We have D⁡(σ)=τ−1​(D⁡(τ′​σ​τ))D(\sigma)=\tau^{-1}(D(\tau^{\prime}\sigma\tau)) and D⁡(σ′′)=τ−1​(D⁡(σ′′​τ))D(\sigma^{\prime\prime})=\tau^{-1}(D(\sigma^{\prime\prime}\tau)) (proof of Lemma 5.4.7). The lemma follows now from the previous case applied to the decomposition τ′​σ​τ=(τ′​σ′)​(σ′′​τ)\tau^{\prime}\sigma\tau=(\tau^{\prime}\sigma^{\prime})(\sigma^{\prime\prime}\tau). ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2