Abstract. This article explains basic constructions and results on group algebras and their cohomology, starting from the point of view of commutative algebra. It provides the background necessary for a novice in this subject to begin reading Dave Benson's article in this volume.
The available accounts of group algebras and group cohomology [Benson 1991a; 1991b; Brown 1982; Evens 1991] are all written for the mathematician on the street. This one is written for commutative algebraists by one of their own. There is a point to such an exercise: though group algebras are typically noncommutative, module theory over them shares many properties with that over commutative rings. Thus, an exposition that draws on these parallels could benefit an algebraist familiar with the commutative world. However, such an endeavour is not without its pitfalls, for often there are subtle differences between the two situations. I have tried to draw attention to similarities and to discrepancies between the two subjects in a series of commentaries on the text that appear under the rubric Ramble.
The approach I have adopted toward group cohomology is entirely algebraic. However, one cannot go too far into it without some familiarity with algebraic topology. To gain an appreciation of the connections between these two subjects, and for a history of group cohomology, one might read [Benson and Kropholler 1995; Mac Lane 1978].
In preparing this article, I had the good fortune of having innumerable ‘chalk-and-board’ conversations with Lucho Avramov and Dave Benson.