Algebraic cycles are a central theme in algebraic geometry, appearing in places such as Abel’s Theorem, The Riemann-Roch Theorem, enumerative geometry, higher K-theory, motivic cohomology, and the Hodge conjecture. In this course we develop some basic ideas, and review some of these applications. For more information see: http://www.math.titech.ac.jp/~shanekelly/Cycles2018-19WS.html
(1) Obtain overall knowledge on basics of algebraic cycle theories, such as Chow groups
(2) Understand the relationship between Chow groups and other theories, such as de Rham cohomology
(3) Attain basic understanding of motivic cohomology
Algebraic cycles, Chow groups, motivic cohomology
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Category of finite correspondances | Details will be provided during each class session. |
Class 2 | Presheaves with transfers | Details will be provided during each class session. |
Class 3 | Motivic cohomology | Details will be provided during each class session. |
Class 4 | Weight one motivic cohomology | Details will be provided during each class session. |
Class 5 | Milnor K-theory | Details will be provided during each class session. |
Class 6 | Étale sheaves with transfers | Details will be provided during each class session. |
Class 7 | Higher Chow groups | Details will be provided during each class session. |
Class 8 | Voevodsky's category of motives | Details will be provided during each class session. |
None required
Murre, Lectures on algebraic cycles and Chow groups
Mazza, Carlo, Vladimir Voevodsky, and Charles A. Weibel. Lecture notes on motivic cohomology. Vol. 2. American Mathematical Soc., 2011.
Course scores are evaluated by homework assignments. Details will be announced during the course.
Basic knowledge of scheme theory (e.g., Hartshorne)
Details TBD