# Lectures on Functor Homology by Vincent Franjou, Antoine Touzé

By Vincent Franjou, Antoine Touzé

This ebook encompasses a sequence of lectures that explores 3 assorted fields within which functor homology (short for homological algebra in functor different types) has lately performed an important function. for every of those functions, the functor point of view presents either crucial insights and new equipment for tackling tricky mathematical problems.

In the lectures through Aurélien Djament, polynomial functors look as coefficients within the homology of limitless households of classical teams, e.g. normal linear teams or symplectic teams, and their stabilization. Djament’s theorem states that this solid homology will be computed utilizing purely the homology with trivial coefficients and the conceivable functor homology. The sequence contains an exciting improvement of Scorichenko’s unpublished results.

The lectures by way of Wilberd van der Kallen result in the answer of the overall cohomological finite iteration challenge, extending Hilbert’s fourteenth challenge and its method to the context of cohomology. the point of interest this is at the cohomology of algebraic teams, or rational cohomology, and the coefficients are Friedlander and Suslin’s strict polynomial functors, a conceptual kind of modules over the Schur algebra.

Roman Mikhailov’s lectures spotlight topological invariants: homoto

py and homology of topological areas, via derived functors of polynomial functors. during this regard the functor framework makes greater use of naturality, permitting it to arrive calculations that stay past the snatch of classical algebraic topology.

Lastly, Antoine Touzé’s introductory direction on homological algebra makes the publication available to graduate scholars new to the field.

The hyperlinks among functor homology and the 3 fields pointed out above supply compelling arguments for pushing the advance of the functor point of view. The lectures during this ebook will supply readers with a consider for functors, and a useful new viewpoint to use to their favorite problems.

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VdK80] Wilberd van der Kallen. Homology stability for linear groups. Invent. , 60(3):269–295, 1980. [Vog82] K. Vogtmann. A Stiefel complex for the orthogonal group of a ﬁeld. Comment. Math. , 57(1):11–21, 1982. [Wei94] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994. fr Progress in Mathematics, Vol. 311, 41–65 c Springer International Publishing Switzerland 2015 Lectures on Bifunctors and Finite Generation of Rational Cohomology Algebras Wilberd van der Kallen Abstract.

Comment. Math. , 58(1):72–85, 1983. Homologie stable des groupes à coeﬃcients polynomiaux [ML98] 39 Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1998. [MvdK02] B. Mirzaii and W. van der Kallen. Homology stability for unitary groups. Doc. , 7:143–166 (electronic), 2002. [Nak60] Minoru Nakaoka. Decomposition theorem for homology groups of symmetric groups. Ann. of Math. (2), 71:16–42, 1960. [Pir00a] Teimuraz Pirashvili.

In this theory one exploits a connection between the rate of growth of a minimal projective resolution and the dimension of a ‘support variety’, which is a subvariety of the spectrum of H even (G, k). The case of ﬁnite group schemes over a ﬁeld (these are group schemes whose coordinate ring is a ﬁnite-dimensional vector space) turned out to be ‘surprisingly elusive’. It was ﬁnally settled by Friedlander and Suslin (1997) [11]. For this they had to invent strict polynomial functors and compute with certain Ext groups in the category of strict polynomial functors.