Recalling prior material. Rings (including noncommutative), ideals, modules and bimodules, exact sequences, infinite direct sums and products, tensor products of modules and rings.
Categories and functors. Morphisms of functors, equivalences. Linear categories and linear functors. Exactness of functors.
Special modules. Projective, injective and flat modules.
Morita Theory. Equivalences of module categories realized as tensor products.
Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
Resolutions. Projective, injective and flat resolutions – existence and uniqueness.
Left and right derived functors. The general theory. Tor and Ext functors.
Applications to commutative algebra. Some local and global theorems, involving $Tor$ and $Ext$ functors. Derived completion and torsion functors.
Sheaf cohomology. A survey of the role of homological algebra in geometry.
Nonabelian cohomology. A survey of classification theorems: Galois cohomology, vector bundles.
Bibliography
R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.
P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.
S. Maclane, “Homology”, Springer, 1994.
J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.
C. Weibel, “An introduction to homological algebra”, Cambridge Univ. Press, 1994.
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
The Stacks Project, an online reference, J.A. de Jong (Editor).
(9) A. Yekutieli, “Derived Categories”, Cambridge Univ. Press, 2019. Free prepublication version.
(10) Course notes, to be uploaded every week to the course web page