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School of Environment and Society
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- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kelly Shane
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Mon5-6()
- Group
- -
- Course number
- MTH.A505
- Credits
- 1
- Academic year
- 2021
- Offered quarter
- 1Q
- Syllabus updated
- 2021/3/19
- Lecture notes updated
- -
- Language used
- English
- Access Index
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Course description and aims
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
Student learning outcomes
(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 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 | Introduction | Details will be provided during each class session. |
| Class 2 | Commutative Algebra I | Details will be provided during each class session. |
| Class 3 | Topology I | Details will be provided during each class session. |
| Class 4 | Homological Algebra I | Details will be provided during each class session. |
| Class 5 | Functoriality I | Details will be provided during each class session. |
| Class 6 | Étale cohomology I | Details will be provided during each class session. |
| Class 7 | Étale cohomology II | Details will be provided during each class session. |
| Class 8 | Galois theory I | 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 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 required
Reference books, course materials, etc.
Course materials are provided during class.
Assessment criteria and methods
Learning achievement is evaluated by reports (100%).
Related courses
- MTH.A301 ： Algebra I
- MTH.A302 ： Algebra II
- MTH.A331 ： Algebra III
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Basic knowledge of scheme theory (e.g., Hartshorne)
