Motivic Cohomology
Motivic cohomology, originated from Deligne, Beilinson and Lichtenbaum and developed by Voevodsky, is a kind of cohomology theory on schemes. It admits comparison with étale cohomology of powers of roots of unity (Beilinson-Lichtenbaum), together with higher Chow groups, and relates to K-theory by Atiyah-Hirzebruch spectral sequence.
In this lecture, we establish the category of motives in which the motivic cohomologies are realized. We explain its relationship with Milnor K-theory and Chow group, as well as the theory of cycle modules. Furthermore, we introduce cancellation theorem, Gysin triangle, projective bundle formula, BB-decomposition and duality.
In this lecture, we establish the category of motives in which the motivic cohomologies are realized. We explain its relationship with Milnor K-theory and Chow group, as well as the theory of cycle modules. Furthermore, we introduce cancellation theorem, Gysin triangle, projective bundle formula, BB-decomposition and duality.
Lecturer
Date
18th February ~ 3rd June, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 09:50 - 12:15 | A3-4-312 | ZOOM 04 | 482 240 1589 | BIMSA |
Prerequisite
Basic algebraic geometry (GTM 52, Chapter 1-3)
Syllabus
1. Intersection Theory
2. Sheaves with Transfers
3. Milnor K-Theory and Cycle Modules
4. Comparison between Milnor K-Theory and Motivic Cohomology
5. Effective Motivic Categories over Smooth Bases
6. Cancellation Theorem
7. Weight One Motivic Cohomology
8. Comparison between Chow Group and Motivic Cohomology
9. Homotopy Purity, Projective Bundle Theorem, BB-Decomposition and Gysin Triangle
10. Spectra, Symmetric Spectra and Stable Motivic Categories
11. Duality
2. Sheaves with Transfers
3. Milnor K-Theory and Cycle Modules
4. Comparison between Milnor K-Theory and Motivic Cohomology
5. Effective Motivic Categories over Smooth Bases
6. Cancellation Theorem
7. Weight One Motivic Cohomology
8. Comparison between Chow Group and Motivic Cohomology
9. Homotopy Purity, Projective Bundle Theorem, BB-Decomposition and Gysin Triangle
10. Spectra, Symmetric Spectra and Stable Motivic Categories
11. Duality
Reference
C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on Motivic Cohomology, American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006).
M. Rost, Chow Groups with Coefficients, Documenta Mathematica (1996), Volume: 1, page 319-393
M. Rost, Chow Groups with Coefficients, Documenta Mathematica (1996), Volume: 1, page 319-393
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
No
Language
Chinese
Lecturer Intro
Nanjun Yang got his doctor and master degree in University of Grenoble-Alpes, advised by Jean Fasel, and bachelor degree in Beihang University. Then he became a postdoc in YMSC. Currently he is a assistant professor in BIMSA. His research interest is the Chow-Witt group of algebraic varieties, with publications on journals such as Camb. J. Math and Ann. K-Theory.