2021 Advanced topics in Algebra F1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kelly Shane 
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Mon5-6()  
Group
-
Course number
MTH.A506
Credits
1
Academic year
2021
Offered quarter
2Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
English
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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 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.

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)

Unspecified.

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
  • MTH.A501 : Advanced topics in Algebra E

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge of scheme theory (e.g., Hartshorne)

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