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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yoshikawa Shou
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- -
- Group
- -
- Course number
- MTH.A403
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 3Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
Singularities in positive characteristic is useful for algebraic geometry in all characteristic not only in positive characteristic. The aim of this course together with "Advanced topics in Algebra D" is to introduce the basic notion of Frobenius-regularity with a view towards both classical and modern applications.
Student learning outcomes
Students are expected to understand the basic notion of Frobenius regularity and quasi-Frobenius-regularity. Looking through concrete examples and applications, students get acquainted with the fundamental importance of singularities in positive characteristic in current research.
Keywords
Commutative ring, Singularities, Frobenius morphisms, Witt ring.
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Standard lecture course
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Commutative ring theory in positive characteristic | Details will be provided during each class session |
| Class 2 | Frobenius morphisms and Kunz's theorem | Details will be provided during each class session |
| Class 3 | Frobenius splitting | Details will be provided during each class session |
| Class 4 | Frobenius regularity | Details will be provided during each class session |
| Class 5 | Fedder's criterion | Details will be provided during each class session |
| Class 6 | Test ideal | Details will be provided during each class session |
| Class 7 | Applications for Frobenius regularity | Details will be provided during each class session |
Out-of-Class Study Time (Preparation and Review)
To enhance effective learning, students are encouraged to explore references provided in lectures and other materials.
Textbook(s)
None required.
Reference books, course materials, etc.
Matsumura, Hideyuki, Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, 1986.
Karl Schwede, Kevin Tucker, A survey of test ideals, arXiv:1104.2000, 2000.
Assessment criteria and methods
Course scores are evaluated by homework assignments. Details will be announced during the course.
Related courses
- MTH.A404 ： Advanced topics in Algebra D
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic undergraduate algebra in particular, commutative ring theory.
Other
None in particular.
