2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics

Special lectures on current topics in Mathematics B

Academic unit or major: Graduate major in Mathematics
Instructor(s): Shuji Saito / Yuri Yatagawa
Class Format: Lecture (Face-to-face)
Media-enhanced courses: -
Day of week/Period (Classrooms): Intensive (Main Bldg,2Fl,Room No.201)
Class: -
Course Code: MTH.E632
Number of credits: 200
Course offered: 2025
Offered quarter: 3Q
Syllabus updated: Sep 29, 2025
Language: Japanese

Syllabus

Course overview and goals

Étale cohomology theory plays a fundamental role in arithmetic geometry. Tame topology is a refinement of the étale topology, defined by restricting the family of étale coverings of a scheme X to those with tame ramification along the boundary of a compactification of X, or to those with tame ramification along the special fiber of a model of X over the ring of integers (independently of the choice of a compactification or a model). This lecture will explain the tame topology, its cohomology theory, and its applications. The following topics will be covered:
1. Review of Grothendieck topology and valuation fields, followed by the definition of the tame topology.
2. Explanation of the fundamental theory of tame cohomology: Determination of the local rings of the tame topology. A general construction method of tame sheaves and its examples. Various methods for computing tame cohomology.
3. GAGA: A comparison theorem between the tame cohomology of a scheme X over a complete discrete valuation field and the cohomology of the associated rigid analytic space X^an.
4. As an application of 3, construction and properties of a canonical integral structure of the Hodge cohomology of schemes over a complete discrete valuation field.

Course description and aims

Understanding of tame cohomology theory and its applications.

Keywords

Tame topology, tame cohomology

Competencies

Specialist skills
Intercultural skills
Communication skills
Critical thinking skills
Practical and/or problem-solving skills

Class flow

The course will be conducted in the usual lecture format. Reports will be assigned as necessary.

Course schedule/Objectives

	Course schedule	Objectives
Class 1	1. Review of Grothendieck topology and valuation fields, followed by the definition of the tame topology. 2. Explanation of the basic theory of tame cohomology (I): Determination of the local ring of the tame topology. A general construction method of tame sheaves and its examples.. 3. Explanation of the basic theory of tame cohomology (II): Various methods for computing tame cohomology. 4. GAGA: A comparison theorem between the tame cohomology of a scheme X over a complete discrete valuation field and the cohomology of the associated rigid analytic space X^an. ５. As an application of４, construction and properties of a canonical integral structure of the Hodge cohomology of schemes over a complete discrete valuation field.	Details will be provided during each class session

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to course material.

Textbook(s)

None.

Reference books, course materials, etc.

None in particular.

Evaluation methods and criteria

Assignments (100%)

Related courses

MTH.A401 ： Advanced topics in Algebra A
MTH.A402 ： Advanced topics in Algebra B
MTH.A501 ： Advanced topics in Algebra E
MTH.A502 ： Advanced topics in Algebra F

Prerequisites

Students are expected to have reviewed Chapter 3 of Hartshorne (especially derived functors and sheaf cohomology), and basic category theory, including adjoint functors and Kan extensions.