2017 Advanced topics in Algebra E1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Ma Shohei  Somekawa Mutsuro 
Course component(s)
Lecture
Day/Period(Room No.)
Mon5-6(H116)  
Group
-
Course number
MTH.A505
Credits
1
Academic year
2017
Offered quarter
1Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

The main topics of this course are the concepts and properties of étale cohomology, often used for the arithmetic geometry. After reviewing Galois theory for fields, students will study the definition and properties of Galois cohomology, and specific examples. The étale cohomology of schemes for fields is Galois cohomology. Étale cohomology is a natural extension of Galois cohomology. Students will then review the basics of schemes and sheaves, studying étale morphism and Grothendieck topology. Using that, the instructor will define étale cohomology, and cover its properties. Specific examples will be investigated. This course is followed by Advanced Topics in Algebra F.
Étale cohomology is a basic, commonly used tool in the arithmetic geometry. Students in this course will gain an understanding of étale cohomology, and accurately describe étale cohomology for algebraic curves over finite fields in particular.

Student learning outcomes

By the end of this course, students will be able to:
1) Understand the definition and some of the basic properties of étale cohomologies,
2) Understand the relation between étale cohomologies, Galois cohomologies and Zariski cohomologies,
3) Calculation of low-dimensional Galois cohomologies,
4) Calculation of low-dimensional étale cohomologies.

Keywords

Galois cohomology, scheme, Zariski cohomology, étale morphism, Grothendieck topology, étale cohomology

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 the abelian category and the general theory of cohomologies Details will be provided during each class session.
Class 3 review of the infinite dimensional Galois theory and the definition and some properties of Galois cohomologies Details will be provided during each class session.
Class 4 the cohomoloy theory with coefficients in a sheaf on the Grothendieck topology Details will be provided during each class session.
Class 5 review of schemes and the zariski topology Details will be provided during each class session.
Class 6 étale morphisms Details will be provided during each class session.
Class 7 the definition and some of the basic properties of étale cohomologies Details will be provided during each class session.
Class 8 étale cohomologies of fields Details will be provided during each class session.

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

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

Students must have successfully completed Algebra I, Algebra II and Algebra III;
or, they must have equivalent knowledge.

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