2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra F1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shane Kelly
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon
- Class
- -
- Course Code
- MTH.A506
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
Motivated by Weil's beautiful conjectures on zeta functions counting points on varieties over finite fields, étale cohomology is a theory generalising singular cohomology of complex algebraic varieties. In the first half we give an introduction to the classical theory of étale cohomology. In the second half, we will discuss Bhatt-Scholze's pro-étale topology. For more information see: http://www.math.titech.ac.jp/~shanekelly/EtaleCohomology2019SS.html
Course description and aims
(1) Obtain overall knowledge on basics in étale cohomology
(2) Understand the relationship between étale topology and Galois theory
(3) Attain understanding of possible applications of étale topology
Keywords
Étale cohomology, homological algebra, Galois theory
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The pro-étale topology | Details will be provided during each class session. |
| Class 2 | Commutative algebra II | Details will be provided during each class session. |
| Class 3 | Homological algebra II | Details will be provided during each class session. |
| Class 4 | Homological algebra III | Details will be provided during each class session. |
| Class 5 | Topology II | Details will be provided during each class session. |
| Class 6 | Functoriality II | Details will be provided during each class session. |
| Class 7 | Galois theory II | Details will be provided during each class session. |
| Class 8 | Review | 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)
Unspecified.
Reference books, course materials, etc.
Course materials are provided during class.
Evaluation methods and criteria
Learning achievement is evaluated by reports(100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A331 ： Algebra III
- MTH.A501 ： Advanced topics in Algebra E
Prerequisites
Basic knowledge of scheme theory (e.g., Hartshorne)