## 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.
http://www.cmi.ac.in/~mkummini/index.html

**Office**: Room FR-3.

**Office Hours**: Mondays Noon-1pm

**Important information**:
- Moodle.
I'll add this information and details on Moodle later;
After you log in, search for `Commutative Algebra' under
`Mathematics /Aug - Dec 2013'. Assignments and grades
will be posted in Moodle.

**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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Last modified:
Wed Jul 10 07:00:58 IST 2013