2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra G
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shou Yoshikawa
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.A503
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 3Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course explains the relationship between the global properties of projective varieties and the local properties of graded rings. Such a correspondence is often called the cone correspondence. We also introduce various applications, including Horrocks’ splitting theorem.
In this course, together with Advanced Topics in Algebra H, we begin by reviewing fundamental concepts in algebraic geometry, such as affine schemes, projective schemes, and dualizing complexes. We then present the basic theory of the cone correspondence. Through this, we examine how graded rings can be used in the study of projective varieties.
Course description and aims
- To be able to explain the definition and examples of dualizing complex.
- To be able to state Matlis duality.
- To be able to explain the definition and examples of affine scheme and projective scheme.
- To be able to explain the sketch of the proof of Horrocks splitting theorem.
Keywords
Dualizing complex, graded ring, scheme, Horrocks splitting theorem.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course. Assignments will be given during class sessions.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Affine schemes |
Details will be provided during each class session. |
| Class 2 | Projective spaces I |
Details will be provided during each class session. |
| Class 3 | Projective spaces II |
Details will be provided during each class session. |
| Class 4 | Dualizing module on projective spaces I |
Details will be provided during each class session. |
| Class 5 | Dualizing module on projective spaces II |
Details will be provided during each class session. |
| Class 6 | Horrocks splitting theorem I |
Details will be provided during each class session. |
| Class 7 | Horrocks splitting theorem II |
Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 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 in particular.
Reference books, course materials, etc.
・Stacks Project. Tag 08XG Dualizing Complexes, https://stacks.math.columbia.edu/tag/08XG.
・R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York–Heidelberg, 1977.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A507 ： Advanced topics in Algebra G1
Prerequisites
Basic knowledge on algebra is expected. Note: Some basic facts from the derived category are used.
Other
None in particular.