Commutative Algebra

Lecture hours/location: Lecture Hall 1, Mondays 15:30-16:45 and Fridays 10:30-11:45
Instructor: Manoj Kummini. Email: mkummini AT cmi DOT ac DOT in.
Office: Room FR-3.
Office Hours: Mondays Noon-1pm

Important information: Course description: This is the MSc 2nd year core course in commutative algebra. Textbook: D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer. Additional references: J.-P. Serre, Local algebra, Springer. M. F. Atiyah and I. G. Macdonald, An introduction to commutative algebra, Addison-Wesley. H. Matsumura, Commutative Ring Theory, Cambridge. Tentative list of topics to be covered: Numbers refer to the sections in Eisenbud's book. We will take some proofs from the other references listed above. Rings: definition, examples incl. rings of functions, polynomial rings as rings of functions, PID, UFD. Ideals, quotient rings. Homomorphisms Affine rings. (Chapter 0) Prime ideals, maximal ideals. Spectrum. Nilradical, Jacobson radical, NAK. (Chapter 0, Section 4.1) Modules: Hom and tensor: Projective, injective and flat modules. (Chapter 0, Section 2.2) Localization, local rings, (Chapter 2) Noetherian rings, Hilbert's basis theorem (Section 1.4). Primary decomposition and associated primes (Sections 3.1-3.8). Artinian rings (Section 2.4) Integral extensions, going up, down. (Sections 4.2, 4.4, 13.1) Noether's Normalization Lemma (Section 13.1), Hilbert's Nullstellensatz (Sections 4.5, 13.2) Dimension theory (Sections 10.1, 12.1) Further topics (if time permits): Regular local rings. Free resolutions, Koszul complexes, Auslander-Buchsbaum-Serre characterization of RLRs. (Chapter 19)

Grading: Assignments (30%), Midterm (30%), final examination (30%) and presentation/oral examination (10%).

Prerequisites: This is open to MSc students and 3rd year BSc students who have taken a course in basic ring theory. If you are a 2nd year BSc student who is planning to take this course concurrently with Algebra III, talk to me before you enrol.

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