Email : schapira at math dot jussieu dot fr
The first part of the course will be devoted to homological algebra, introducing derived functors and giving examples and tools to calculate them. Derived categories will be mentionned, but not constructed in detail.
The second part will expose the theory of abelian sheaves on topological spaces (with a glance at Grothendieck topologies) and their operations, hom and tens, direct and inverse images.
As an application, we will mention the De Rham and Dolbeault cohomologies of real and complex manifolds and integration on complex manifolds (Leray-Grothendieck residues).
The course will be partially based on the Notes [5]. A detailed exposition of derived categories and sheaves is given in [1] and [2]. The references [3] and [4] go much beyond the scope of this course.
[1] S.I. Gelfand and Yu.I. Manin, Methods of homological algebra, Springer (1996)
[2] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Ch. 1 & 2, Grundlehren der Math. Wiss. 292 Springer-Verlag (1990)
[3] Categories and Sheaves, Grundlehren der Math. Wiss. 332 Springer-Verlag (2005)
[4] S-G-A 4, Sém. Géom. Algébrique (1963-64) by M. Artin, A. Grothendieck and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 269, 270, 305, Springer-Verlag (1972/73)
[5] P. Schapira,
Categories and Homological Algebra, Abelian Sheaves,
http://www.math.jussieu.fr/~schapira/polycopies/