MTH 423 (Spring 2019) (Commutative algebra)

This is an undergraduate course on commutative algebra. Please read the course template for text-books and other important information about the course.

Grading :

  1. Mid-Semester Exams-30%
  2. End-Semester Exams-30%.
  3. Short tests-20%
  4. Class tests (2):-10%+10%.

Please note that there will not be any repeat class tests.

Recommended Reading:

    1. Commutative algebra: Atiyah and MacDonald (1994) Westview Press
    2. Commutative algebra: Gopalkrishnan (1984) Oxonian Press
    3. Undergraduate commutative algebra: Miles Reid.

Prerequisites: Vector Spaces, Rings and Modules (MTH 320)

Class timing:

Monday-5.30-6.30pm

Wednesday-9. 40-10.40am

Friday-2-3pm

Class room

LHC 304

Week 1

02/01/2019: Introduction to the course

04/01/2019:Ring homomorphism, ideals, quotients, zero divisors, nilpotents and units.

Week 2

07/01/2019: Prime ideals, maximal ideals ,existence of maximal ideals (Spectrum of a ring).

09/01/2019:Nilradical and Jacobson radical.

11/01/2019:Tutorial

Week 3

14/01/2019:Some properties of Ideals like Prime avoidance lemma, Chinese Remainder theorem and

colon of an ideal.

16/01/2019: Basic constructions for modules, like direct sums, direct products,

18/01/2019: Short test 1.

Week 4

21/01/2019:Short exact sequences,

23/01/2019:Hom functor

25/01/2019: Tutorial

Week 5

28/01/2019:Cayley-Hamilton theorem for endomorphisms of a finitely generated R- module , Nakayama's Lemma.

30/01/2019: Projective modules

01/02/2019: Class test 1.

Week 6

04/02/2019:Tensor products

06/02/2019: Flat modules.

08/02/2019: Short test II

Week 7

11/02/2019: Some more properties of projective modules.

13/02/2019: Tutorial.

15/02/2019: No class, please prepare for mid-sem exam.

Weak M

Mid-Semester examination, no class

Mid-sem on 25/02/2019

Week 8

25/02/2019: Mid-Sem exam

27/02/2019: Discussion on mid-sem exam; revision.

01/03/2019: Localization, important properties of localization like localization is an exact functor

Week 9

04/03/2019: Holiday

06/03/2019: Other important properties of localization like it commutes with tensor products, correspondence of prime ideals under localization.

08/03/2019: Tutorial.

Week 10

11/03/2019: No class; instructor will be at the International conference on number theory at the IISER, Tiruvananthapuram.

13/03/2019: Noetherian rings

15/03/2019: Short test 3.

Week 11

18/03/2019:Hilbert basis theorem and it's applications.

20/03/2019: Noetherian ring (properties)

22/03/2019: Tutorial.

Week 12

25/03/2019: Integral extensions

27/03/2019: Going up theorem.

29/03/2019:Class test II

Week 13

01/04/2019: Going down theorem.

03/04/2019: Applications of Integral extensions including Noether Normalization Lemma.

05/04/2019:Tutorial.

Week 14

08/04/2019: Primary decomposition.

10/04/2019: First uniqueness theorem of primary decomposition, Revision.

12/04/2019: Short test 4.

End of the course. The material taught in week 15 is for your intellectual satisfaction.

Week 15

These topics are not part of end- semester examination syllabus.

15/04/2019: Artin rings, revision.

17/04/2019: Holiday (Mahavir Jayanti)

19/04/2019: Holiday (Good Friday).

Week 16

End sem exam on 22/04/2019.

KINDLY MEET ME BEFORE 26TH APRIL, 2019 TO SEE END-SEM EXAM PAPER.