ScalingStacks

Let TT and T′T^{\prime} be two finite subsets of ZZ such that |f⁡(T)|=|T||f(T)|=|T| and |f⁡(T′)|=|T′||f(T^{\prime})|=|T^{\prime}|. Put T̊=T∖(T∩ξ~​({−n,…,−1})CLOSE\mathring{T}=T\setminus(T\cap\tilde{\xi}(\{-n,\ldots,-1\}) and T̊′=T′∖(T′∩ξ~​({−n,…,−1})CLOSE\mathring{T}^{\prime}=T^{\prime}\setminus(T^{\prime}\cap\tilde{\xi}(\{-n,\ldots,-1\}). There is a commutative diagram

(8.1.6) Hom𝒮∙​(Z)⁡(T,T′)\textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T,T^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}κn\scriptstyle{\kappa_{n}}f\scriptstyle{f}Hom𝒮∙​(Z)⁡(T̊,T̊′)\textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(\mathring{T},\mathring{T}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}f\scriptstyle{f}Hom𝒮∙​(Z¯)⁡(f⁡(T),f⁡(T′))\textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(\bar{Z})}(f(T),f(T^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}κn\scriptstyle{\kappa_{n}}Hom𝒮∙​(Z¯)⁡(f⁡(T̊),f⁡(T̊′))\textstyle{\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(\bar{Z})}(f(\mathring{T}),f(\mathring{T}^{\prime}))}

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2