2023 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Algebra A
- Academic unit or major
- Mathematics
- Instructor(s)
- Kazuma Shimomoto
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu (M-101(H116))
- Class
- -
- Course Code
- ZUA.A331
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
The object of the lectures is to learn commutative ring theory and its applications. The knowledge of ring theory is not only the basis of algebraic geometry and number theory, but it becomes an indispensable language for learning other branches of mathematics. We begin to explain local cohomology and basic properties, regular sequnces, Cohen-Macaulay rings, as well as their geometric meanings.
Course description and aims
1. Understand local cohomology modules and its relation with regular sequences
2. Learn how to compute local cohomology modules
3. Understand Cohen-Macaulay rings via local cohomology
4. Construct examples of Cohen-Macaulay rings
Keywords
injective module, projective module, regular sequence, local cohomology, Cohen-Macaulay ring
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | We will discuss the following topics in the lectures. (1) injective module and injective resolution (2) Ext-modules and regular sequnece (3) definition of local cohomology modules (4) local cohomology with its connection to Cohen-Macaulay rings (5) Gorenstein ring (6) vanishing theorem and local duality theorem (7) applications to algebraic geometry | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 30 minutes preparing for class and another 30 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required.
Reference books, course materials, etc.
「Cohen-Macaulay Rings」：W.Bruns and J.Herzog
「Commutative Ring Theory」：H. Matsumura
「Introduction to Commutative Algebra and Algebraic Geometry」：E. kunz
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
Prerequisites
Basic knowledge of some abstract algebra, including rings and modules, is preferable.