This book features a series of lectures that explores three different fields in which functor homology (short for homological algebra in functor categories) has recently played a significant role. For each of these applications, the functor viewpoint provides both essential insights and new methods for tackling difficult mathematical problems.
In the lectures by Aurelien Djament, polynomial functors appear as coefficients in the homology of infinite families of classical groups, e.g. general linear groups or symplectic groups, and their stabilization. Djament's theorem states that this stable homology can be computed using only the homology with trivial coefficients and the manageable functor homology. The series includes an intriguing development of Scorichenko's unpublished results.
The lectures by Wilberd van der Kallen lead to the solution of the general cohomological finite generation problem, extending Hilbert's fourteenth problem and its solution to the context of cohomology. The focus here is on the cohomology of algebraic groups, or rational cohomology, and the coefficients are Friedlander and Suslin's strict polynomial functors, a conceptual form of modules over the Schur algebra.
Roman Mikhailov's lectures highlight topological invariants: homoto
py and homology of topological spaces, through derived functors of polynomial functors. In this regard the functor framework makes better use of naturality, allowing it to reach calculations that remain beyond the grasp of classical algebraic topology.
Lastly, Antoine Touze's introductory course on homological algebra makes the book accessible to graduate students new to the field.
The links between functor homology and the three fields mentioned above offer compelling arguments for pushing the development of the functor viewpoint. The lectures in this book will provide readers with a feel for functors, and a valuable new perspective to apply to their favourite problems.
Aurelien Djament A. Djament is a CNRS researcher at the LMJL in Nantes. He studies functors categories and homological stability of infinite families of classical groups. Wilberd van der Kallen W. van der Kallen is a retired professor at the Mathematical Institute of Utrecht University. He is interested in representations of algebraic groups. Roman V. Mikhailov R. Mikhailkov is a researcher at the Steklov Institute of Mathematics in St-Petersburg. His interests include group theory, topology, category theory, and algebraic K-theory. Antoine Touze A. Touze holds a CNRS/Universite Paris 13 chair as a Maitre de Conferences at the LAGA. He is interested in algebraic groups, homological algebra, and algebraic topology.
Introduction.- A. Djament: Homologie stable des groupes a coefficients polynomiaux.- W. van der Kallen: Lectures on Bifunctors and Finite Generation of Rational Cohomology Algebras.- R. Mikhailov: Polynomial Functors and Homotopy Theory.- A. Touze: Prerequisites of Homological Algebra.