### 2019　Special courses on advanced topics in Mathematics H

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Mathematics
Instructor(s)
Sato Kanetomo
Course component(s)
Lecture
Day/Period(Room No.)
Intensive ()
Group
-
Course number
ZUA.E342
Credits
2
2019
Offered quarter
2Q
Syllabus updated
2019/6/17
Lecture notes updated
-
Language used
English
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### Course description and aims

Chow group of algebraic varieties (more generally, of noetherian schemes) include the well-known classical invariants like the ideal class group of algebraic integer rings and the divisor class group of Riemannian surfaces. We will start this course with overview and basic materials on algebraic cycles, and then introduce the Brauer-Manin pairing, which is an important tool to study the Chow groups of 0-cycles of varieties over p-adic fields to explain a certain non-degeneracy fact on this pairing. Moreover, we will introduce an arithmetic cohomology theory of schemes over integer rings, and expain that this non-degeneracy is closely related with the surjectivity of a cycle class map with values in this arithmetic cohomology groups.

### Student learning outcomes

・Understand the definition of Chow groups
・Understand the relation between Chow groups of 0-cycles and Brauer groups of varieties over p-adic fields
・Understand the relation between the surjectivity of cycle class map of schemes over integer rings with well-known problems on cycles.

### Keywords

algebraic variety, algebraic cycle, Chow group, Brauer group, Brauer-Manin pairing, cycle class map

### Competencies that will be developed

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

### Class flow

This is an intensive lecture course. There will be some assignments.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Overview, and some basics on algebraic cycles Details will be provided during each class session
Class 2 rational equivalence and Chow's moving lemma Details will be provided during each class session
Class 3 Chow groups of 0-cycles and Albanese mappings Details will be provided during each class session
Class 4 relation between the Chow groups and the Grothendieck group 1 Details will be provided during each class session
Class 5 relation between the Chow groups and the Grothendieck group 2 Details will be provided during each class session
Class 6 Brauer group of a field and Galois cohomology 1 Details will be provided during each class session
Class 7 Brauer group of a field and Galois cohomology 2 Details will be provided during each class session
Class 8 Brauer gorup of a scheme and étale cohomoloy 1 Details will be provided during each class session
Class 9 Brauer gorup of a scheme and étale cohomoloy 2 Details will be provided during each class session
Class 10 Lichtenbaum-Manin pairing of varieties over p-adic fields 1 Details will be provided during each class session
Class 11 Lichtenbaum-Manin pairing of varieties over p-adic fields 2 Details will be provided during each class session
Class 12 Lichtenbaum-Manin pairing of varieties over p-adic fields 3 Details will be provided during each class session
Class 13 cycle class maps of schemes over integer rings 1 Details will be provided during each class session
Class 14 cycle class maps of schemes over integer rings 2 Details will be provided during each class session
Class 15 cycle class maps of schemes over integer rings 3 Details will be provided during each class session

None required

### Reference books, course materials, etc.

Hartshorne, R.: Algebraic Geometry, (Graduate Texts in Math. 52), Springer 1977
Saito, S., Sato, K.: Algebraic cycles and Étale cohomology, (Springer Modern Math. Series 17), Maruzen 2012 (in Japanese)

### Assessment criteria and methods

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

Basic knowledge on algebra is expected