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2018　Advanced courses in Algebra D

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Academic unit or major
Mathematics
Instructor(s)
Kelly Shane 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Thr5-6(H137)  
Group
-
Course number
ZUA.A334
Credits
1
Academic year
2018
Offered quarter
4Q
Syllabus updated
2018/9/17
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/EtaleCohomology2018-19WS.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.
Class 2 Commutative algebra II Details will be provided during each class.
Class 3 Homological algebra II Details will be provided during each class.
Class 4 Topology II Details will be provided during each class.
Class 5 Functoriality II Details will be provided during each class.
Class 6 Functoriality III Details will be provided during each class.
Class 7 Functoriality IV Details will be provided during each class.
Class 8 Fundamental group II Details will be provided during each class.

Textbook(s)

None required

Reference books, course materials, etc.

Milne, James S. "Etale cohomology, volume 33 of Princeton Mathematical Series." (1980).
Bhatt, Bhargav, and Peter Scholze. "The pro-\'etale topology for schemes." arXiv preprint arXiv:1309.1198 (2013).

Assessment criteria and methods

Based on the reports with answers of exercise problems presented in the class.

Related courses

  • MTH.A407 ： Advanced topics in Algebra C1
  • MTH.A408 ： Advanced topics in Algebra D1
  • ZUA.A333 ： Advanced courses in Algebra C

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

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

Other

Details TBD

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