Derived Algebraic Geometry

General information

Instructor Shane Kelly

Email shanekelly [at] g.ecc [dot] u-tokyo [dot] ac.jp

Webpage https://www.ms.u-tokyo.ac.jp/~kelly/Course2022-23DAG/2022-23DAG.html

Main References [HTT] Lurie, "Higher Topos Theory" pdf
[DAG] Lurie, "Derived algebraic geometry" pdf
[Toën] Toën, "Derived algebraic geometry" pdf

Other References [SAG] Lurie, "Spectral algebraic geomegry" pdf
[HA] Lurie, "Higher Algebra" pdf

Room 118 Note: The classes will be conducted in hybrid mode. That is, I will lecture in 118 and live-cast to zoom. If you want to the zoom url, send me an email.

Time Thurs(木) 10:25-12:10

Assessment Exercises will be given during the lectures. To pass the course, it is enough to submit solutions to at least one exercise from each lecture (but you are welcome to submit as many solutions as you want).

Please submit the exercise solutions via email by 2th Feb.

日本語でもOKです。

If you have any questions at all about anything to do with the exercises, please write me an email! (There may be small hypothesis errors or typos, so if an exercise seems too difficult let me know.)

Outline (Under construction)

1. Motivation (10月6日)

2022DAG01.pdf

Bezout's Theorem.

2. Homotopy types (10月13日)

2022DAG02.pdf

Topological spaces, simplicial sets, (geometric realisation).

3. Infinity categories I (10月20日)

2022DAG03.pdf

Quasi-categories, simplicial categories, (mapping spaces).

4. Infinity categories II (10月27日)

2022DAG04.pdf

Simplicial model categories, (Yoneda), (localisation).

--- 11月3日文化の日 No lecture 講義がありません ---

5. Colimits I (11月10日)

2022DAG05.pdf

Colimits in quasi-categories, (adjunctions).

6. Colimits II (11月17日)

2022DAG06.pdf

(Derived functors), derived limits, weighted limits.

--- 11月24日勤労感謝の日 No lecture 講義がありません ---

7. Question session (12月1日)

8. Higher algebra I (12月8日)

2022DAG08.pdf

Derived category of a ring, stable ∞-categories, Dold-Kan.

9. Higher algebra II (12月15日)

2022DAG09.pdf

Cotangent complex, morphisms of finite presentation.

10. Higher algebra III (12月22日)

2022DAG10.pdf

Deformations, étale, flat, smooth, morphisms.

--- 12月29日 No lecture 講義がありません ---

11. Higher topoi (1月5日)

2022DAG11.pdf

Topologies, sheaves.

12. Derived schemes (1月12日)

2022DAG12.pdf

13. Quasi-coherent sheaves (1月19日)

2022DAG013.pdf