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Extensions for Finite Chevalley Groups III: Rational and Generic Cohomology

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for generic cohomology which improves on seminal work of Cline, Parshall, Scott and van der Kallen.