Course:MATH600D 17/18-II
| Faculty of Science Department of Mathematics | |
|---|---|
| |
| MATH 600 / D Topics in Algebra | |
| Instructor: | Sujatha Ramdorai |
| Office: | MATH Annex 1201 |
| Class schedule: | Tue Thu<br\>11:00 am - 12:30 pm |
| Classroom: | MATH Annex 1102 |
| Office hours: | |
| Course pages | |
| No pages found. | |
In this course, we plan to cover some parts of Commutative Algebra that are used
in Algebraic Number Theory and Algebraic Geometry. Some of these topics will include
Integral closure, Dedekind domains and Discrete valuation rings. We will then cover some
ground from Homological Algebra, primarily leading to the concept of homological dimension
of Noetherian rings. This will be used to state the Auslander-Buchsbaum Theorem on
regular local rings and provide an overview of key steps in the proof.
The aim is to make the course as self-contained as possible. Students are definitely
expected to be familiar with the material covered in Algebra I (Course no. 501). Some
exposure to Homological Algebra will be helpful, though not strictly necessary. Grades
will be based on class participation, engagement, exercises and lectures. The participants
are expected to give one or two lectures along the course.
There is no prescribed text book. The books listed in the References section will be helpful for the students
Homework
- Homework 1 Due Tuesday, Feb 6th
- Remarks Remarks on Homework 1 Problems
- Homework 2 Due Tuesday, Feb 27th
- Homework 3 Due Thursdaay, March 22nd
Student Lectures
- Projective and Flat Modules and Noetherian Rings First Topic for Presentation
- Second Topic for Presentation
- 3rd and 4th Topics
Lecture Details
| Lecture<br\>Date | Topics Covered | Notes | |
|---|---|---|---|
| 1 | 4/1 |
Integral Extensions | Class 1 notes |
| 2 | 9/1 |
Integral Extensions (contd.) | |
| 3 | 11/1 |
Norms and Traces, Spectrums of Int. Extns | |
| 4 | 16/1 |
Height, Dimension and Localization | |
| 5 | 18/1 |
Projective Dimension, Going Up and Down | |
| 6 | 23/1 |
Going Down (contd.), Flat Extensions | |
| 7 | 25/1 |
||
| 8 | 30/1 |
UFDs | |
| 9 | 1/2 |
Discrete Valuation Rings; Noetherian Modules (Coco's talk) | Coco's talk |
| 10 | 6/2 |
Normal and Dedekind domains | |
| 11 | 8/2 |
Projective and Flat Modules: Coco and Ashvni |
References
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Inc.
- H. Matsumara, Commutative Algebra. W.A. Benjamin.
- E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Springer.
- C. Weibel, An Introduction to Homological Algebra. Cambridge University Press.
- TIFR Pamphlet on Homological Algebra