树与穿孔仿射直线上的${\rm PGL}_2$-挠子的乘积
Products of trees and ${\rm PGL}_2$-torsors over the punctured affine line
AI总结：
本文通过分析树与仿射建筑乘积上的作用，计算了PGL₂在洛朗多项式环上的伽罗瓦上同调，得到了CGP的对应结果，还确定了相关多项式环点群的有限子群，论述自成体系。
AI中文摘要：
本文的目标是计算群$G = \rm{PGL}_2$在洛朗多项式环上的伽罗瓦上同调。该计算通过一种新方法得到了文献[CGP]在该情形下的主要结果，此方法基于对适当几何对象（树的乘积，一般而言是仿射建筑的乘积）上作用的分析，该分析曾在文献[ARR]中用于给出关于多项式环上伽罗瓦上同调的Raghunathan-Ramanathan定理[RR]的新证明。除伽罗瓦上同调外，该方法还能确定相关多项式环点群中的有限子群。考虑到该方法在不同情形下的这些及其他潜在应用（特别是在一般仿射曲线的坐标环上代数群的研究中），我们力求使论述在很大程度上自成体系，以便广大数学工作者理解。
英文摘要：
The goal of this article is to present a computation of the Galois cohomology of the group $G = \mathrm{PGL}_2$ over rings of Laurent polynomials. This computation recovers the main result of \cite{CGP} in this case by a new method, based on the analysis of actions on appropriate geometric objects (products of trees and, in general, of affine buildings), which was already used in \cite{ARR} to give a new proof of the theorem of Raghunathan-Ramanathan \cite{RR} concerning Galois cohomology over polynomial rings. Going beyond Galois cohomology, this method also enables one to determine the finite subgroups in the group of points over relevant polynomial rings. In view of these and other potential applications of the method in different situations (in particular, in the study of algebraic groups over the coordinate rings of general affine curves), we have attempted to make our exposition largely self-contained and accessible to broad mathematical audience.
