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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shimomoto Kazuma
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Thr5-6(M-101(H116))
- Group
- -
- Course number
- MTH.A405
- Credits
- 1
- Academic year
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- 2023/3/20
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
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.
Student learning outcomes
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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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. |
Out-of-Class Study Time (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
Assessment criteria and methods
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 (i.e., required knowledge, skills, courses, etc.)
Basic knowledge of some abstract algebra, including rings and modules, is preferable.
