אלגברה הומולוגית

Description: This is a graduate level course. Undergraduate students can register with my permission.

The prerequisite course is ”Commutative Algebra“ that I gave in the fall semester (or an equivalent course). We will need the following topics from that course: categories and functors; additive and exact functors; free modules; products and coproducts; tensor products of modules and rings.

The pace of the course, and the amount of material covered, will be determined by the background and capabilty of the audience. There will be many examples and exercises. I will upload typed notes after every lecture.

Course Grade: pass/fail grade. Passing the course requires attending all lectures and submitting most of the homework.

Homework: To be assigned every week. Checking will be sporadic.

See the first day handout for more administrative information.

Course Topics:

1. Adjoint functors.
2. Morita Theory.
3. Projective and Injective modules.
4. Complexes of modules.
5. Homotopies and homotopy equivalences.
6. The long exact cohomology sequence.
7. Projective, flat and injective resolutions.
8. Left and right derived functors.
9. Applications of derived functors to commutative algebra.
10. Further applications of derived functors and cohomology.

1. Recalling prior material. Rings (including noncommutative), ideals, modules and bimodules, exact sequences, infinite direct sums and products, tensor products of modules and rings.
2. Categories and functors. Morphisms of functors, equivalences. Linear categories and linear functors. Exactness of functors.
3. Special modules. Projective, injective and flat modules.
4. Morita Theory. Equivalences of module categories realized as tensor products.
5. Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.
6. Resolutions. Projective, injective and flat resolutions – existence and uniqueness.
7. Left and right derived functors. The general theory. Tor and Ext functors.
8. Applications to commutative algebra. Some local and global theorems, involving $Tor$ and $Ext$ functors. Derived completion and torsion functors.
9. Sheaf cohomology. A survey of the role of homological algebra in geometry.
10. Nonabelian cohomology. A survey of classification theorems: Galois cohomology, vector bundles.

Bibliography

1. R. Hartshorne, “Algebraic Geometry”, Springer-Verlag, New-York, 1977.
2. P.J. Hilton and U. Stammbach, “A Course in Homological Algebra”, Springer, 1971.
3. S. Maclane, “Homology”, Springer, 1994.
4. J. Rotman, “An Introduction to Homological Algebra”, Academic Press, 1979.
5. L.R. Rowen, “Ring Theory” (Student Edition), Academic Press, 1991.
6. C. Weibel, “An introduction to homological algebra”, Cambridge Univ. Press, 1994.
7. M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
8. 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