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

Extensions for Finite Chevalley Groups III: Rational and Generic Cohomology