The study of extensions between simple modules is only one aspect of the broader study of cohomology of finite groups, which encodes in a subtle way finer points about the category of KG-modules. This study leads in many directions, some originally motivated by algebraic topology or K-theory rather than representation theory (as in Adem–Milgram). The subject tends to be open-ended, involving diverse methods and many special computations. In our setting cohomology might be thought of as “representation theory by other means”. For groups of Lie type, we continue to emphasize the interaction with cohomology of algebraic groups and their Frobenius kernels, working always in the defining characteristic.
In 14.1 we recall some standard facts about the cohomology of an arbitrary finite group. Then we review briefly in 14.3–14.4 the parallel theory of rational cohomology for algebraic groups and Frobenius kernels, as presented in [RAGS]. Seminal work in the mid-1970s by Cline–Parshall–Scott, especially their joint paper with van der Kallen, relates rational cohomology indirectly to cohomology groups for finite groups of Lie type (14.5). This leads to a number of explicit computations, for example in the further work by Parshall and Friedlander (14.8). The case G = SL(2, q) already illustrates some of the complexities encountered here (14.7).
More recent work by Bendel, Lin, Nakano, Parshall, Pillen, and others approaches in different ways the interplay of cohomology for algebraic groups, Frobenius kernels, and finite groups of Lie type: some of this is discussed in 14.9.