2019 Advanced topics in Algebra E1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kelly Shane 
Course component(s)
Lecture
Day/Period(Room No.)
Mon5-6(H116)  
Group
-
Course number
MTH.A505
Credits
1
Academic year
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
English
Access Index

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

Intercultural skills Communication skills Specialist 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.

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)

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