2016 Advanced topics in Algebra H

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Naito Satoshi  Somekawa Mutsuro 
Course component(s)
Lecture
Day/Period(Room No.)
Mon5-6(H116)  
Group
-
Course number
MTH.A504
Credits
1
Academic year
2016
Offered quarter
4Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

This course deals with reciprocity for algebraic curves over number fields, and provides the basics of algebraic K-groups for schemes. The instructor will first review homotopy theory for topology, then explain the definition and properties of algebraic K-groups, investigating several examples of low-order algebraic K-groups. We will then study the relationship between algebraic K-groups on fields and étale cohomology. Using this, students will construct a reciprocity map for algebraic curves on a number field, and prove reciprocity. This course follows Advanced Topics in Algebra G.
Algebraic K-groups are useful for answering several questions in the geometry of numbers. This course is used for investigating algebraic K-groups and étale cohomology with a reciprocity map of algebraic curves on a number field.

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 the algebraic K-groups of schemes.
2) Understand the structure of the 0th algebraic K-groups and the low-dimensional algebraic K-groups of fields.
3) Understand the relation between the algebraic K-groups and the étale cohomologies of fields.
4) Understand the reciprocity law of an algebraic curve over a number field.

Keywords

homotopy group, algebraic K-group, etale cohomology, algebraic cycle, reciprocity law

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 to the classical class field theory Details will be provided during each class session.
Class 2 the abstract of the algebraic K-theory, and the homotopy groups of categories Details will be provided during each class session.
Class 3 the definition and some of the basic properties of algebraic K-groups of schemes Details will be provided during each class session.
Class 4 the low-dimensional algebraic K-groups, and the algebraic K-groups of fields Details will be provided during each class session.
Class 5 the relation between the algebraic K-group and the étale colomology of fields Details will be provided during each class session.
Class 6 the construction of the reciprocity map Details will be provided during each class session.
Class 7 the reciprocity law for the algebraic curves over local fields Details will be provided during each class session.
Class 8 the reciprocity law for the algebraic curves over number 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.

Related courses

  • MTH.A301 : Algebra I
  • MTH.A302 : Algebra II
  • MTH.A331 : Algebra III
  • MTH.A503 : Advanced topics in Algebra G

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

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

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