Fall 2022
Math 221: Commutative algebra
E-mail: tayou@math.harvard.edu ,
Office: Science Center 238.
Course assistant:
E-mail: ,
Office: Science Center .
Schedule:
- Meeting times: Tuesday-Thursday 09:00 AM-10:15 AM.
- Room: Science center 221.
- First meeting: Thursday, September 1, 2022.
- Office hours: TBD or by appointment, in Science Center 238.
Syllabus:
This class is a graduate level course in Commutative Algebra, also aimed at undergraduates.
Commutative Algebra lies at the foundation of many active research areas in mathematics like Algebraic Geometry, Number Theory and Algebraic Topology. It is also an important subject in itself. The different topics we will cover in this class will give a taste of these different directions.
We will start by studying rings and ideals, associated prime ideals and primary decomposition in Noetherian rings. We will then study dimension theory and briefly touch upon Zariski topology. Then we will move to study integral extensions: going-up, going-down and Noether normalization theorems. The last part of the class will cover valuation rings, graded rings, Hilbert polynomials and homological methods in commutative algebra.
Recommended books:
The main references will be:
- David Eisenbud, "Commutative algebra with a view towards algebraic geometry".
- Atiyah-Macdonald, "Commutative algebra".
- Antoine Chambert-Loir, "(Mostly) Commutative Algebra".
For more advanced reading and exercises, I recommend the following book:
- Hideyuki Matsumura, "Commutative algebra".
Prerequisites:
Math 122-123 (Algebra I-II) or equivalent.
Grading:
Homework will be assigned weekly, on Thursday and will be due the following Thursday. The solutions can either be scanned or typed and uploaded on Canvas. Homework will count for at least 80% of the final grade. Late homework can only be accepted under special circumstances. Collaborative work on homework is accepted but you must write your own solution as well as the names of the collaborators. There will be no final exam but rather a final project (a reading on a topic related to the class), which together with participation will count for 20% of the grade. The Canvas webpage will follow closely the advancement of the class.