##
A COURSE ON COMMUTATIVE ALGEBRA FOR ALGEBRAIC
GEOMETERS

KAPIL HARI PARANJAPE

### 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.

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