### 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
2016
Offered quarter
4Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
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### 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.

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.