About the course

The aim is to introduce the fundamentals of commutative algebra as applicable in other areas; notably algebraic geometry and algebraic number theory.

Books. Auditors are encouraged to examine the following books:

• Atiyah and McDonald: Commutative Algebra
• David Eisenbud: Commutative Algebra
• J. P. Serre: Algebre Locale (Local Algebra)
• O. Zariski and P. Samuel: Commutative Algebra

Topics. The following topics are to be lectured; approximately one topic per week.

1. Finite rings and rings that are finite dimensional over a field. Artinian Rings. Hensel’s lemma.
2. Polynomial rings and their ideals. Hilbert’s basis theorem and Nullstensatz. Notherian Rings and Modules.
3. Associated primes, minimal primes and primary decomposition.
4. Integral extensions. Noether normalisation. Proof of Nullstellensatz. Dimension and transcendence degree.
5. Localisation, gradation and Completion. Revisit earlier theorems for these rings.
6. Filtrations. Artin-Rees Lemma. Krull Intersection Theorem.
7. Tangent vectors and derivations. Tensors and Tensor Products. Module of derivations. Differentials.
8. Categorical constructions. Tensor and Hom and their exact-ness properties. Basic homological algebra.
9. Flatness and fibre-bundles. Generic flat-ness. Elmination theory.
10. Dimension defined in different ways. Hilbert-Samuel polynomial. Height of ideals, number of generators. Krull’s principal ideal theorem.
11. The Koszul complex and minimal resolutions. Depth vs. dimension.
12. Regular local rings. Smoothness. Etale-ness. Hensel’s lemma revisited.
13. Connections with complex analysis and number theory. Counting points. The statement of Weil’s conjectures.

Grading. There will be one assignment per topic. This will carry 5 marks. The end-semester examination will carry 35 marks.