Rational Representations, The Steenrod Algebra and Functor Homology

These notes represent a sequel to a series of lectures delivered in Nantes, December 12-15, 2001 for Société Mathématique de France’s “État de la Recherche” session. They contain the following five articles: T. Pirashvili: Introduction to functor homology; E. M. Friedlander: Lectures on the cohomology of finite group schemes; L. Schwartz: Algèbre de Steenrod, modules instables et foncteur polynomiaux; L. Schwartz: L’algèbre de Steenrod et topologie and V. Franjou & T. Pirashvili: Stable K-theory is bifunctor homology (after A. Scorichenko). The introduction (written by V. Franjou) together with the first paper by T. Pirashvili introduce a reader into the domain of problems under consideration and make him/her familiar with necessary notions. Using polynomial functors introduced in the first article, E. M. Friedlander investigates in his article the cohomology of finite group schemes. The main result states that the cohomology of a finite group scheme is finitely generated. The first article by L. Schwartz presents a completely algebraic description of the Steenrod algebra (over any prime) and deals with unstable modules over this algebra. Again the polynomial functors play an important role here. The second article by L. Schwartz is devoted to the role of the Steenrod algebra in the topological framework. It is only a short note. The last article by V. Franjou & T. Pirashvili brings a result due to A. Scorichenko (with a proof) which shows that stable K-theory is functor homology. The whole book represents a nice introduction to the circle of problems described above. The articles contain the classical results as well as the most recent ones. They are all very well written, and I think that everybody who desires to understand them and is ready to devote a necessary effort, will finally understand them. It is a relatively short but excellent book.

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