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Lectures on Functor Homology (Hardcover, 1st ed. 2015)
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Lectures on Functor Homology (Hardcover, 1st ed. 2015)
Series: Progress in Mathematics, 311
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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Scorichenko's unpublished results.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.
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