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/Course2025DAG/2025DAG.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 Graduate School of Mathmatical Science Bldg. , Room 118.
Time Wed(水) 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 of the first 10 lectures (so at least 10 exercises, but you are welcome to submit as many solutions as you want).

Outline

1. Introduction (4月9日)

2025DAG01.pdf

We use Bezout's Theorem to argue that moving from sets to homotopy types is a natural progression, analogous to fields ⇝ algebraically closed fields; affine varieties ⇝ projective varieties; varities ⇝ schemes.

--- 4月16日 No lecture 講義がありません ---

(April 14-18 Workshop Arithmetic France-Japan - 日本 x フランス 数論幾何学 2025)

2. Homotopy types (4月23日)

2025DAG02.pdf

We present two models for the theory of homotopy types: simplicial sets (or rather Kan complexes) and topological spaces (or rather CW complexes). Most importantly, we define what is a space and a equivalence.

--- 4月30日 No lecture 講義がありません ---

3. Categories (5月7日)

2025DAG03.pdf

We present two models for the theory of infinity categories: quasi-categories and simplicial categories. We define the quasi-categories of spaces.

4. Limits and colimits (5月14日)

2025DAG04.pdf

We discuss weighted limits in simplicial categories, and limits in quasi-categories. We relate these to each other, at least in the case of the simplicial/quasi- category of Kan complexes, that is, the ∞-category of homotopy types.

5. Sheaves (5月21日)

2025DAG05.pdf
Notes on the [HTT] proof of sheafification
Topologies vs. pretopologies

We discuss the notion of sheaf in the classical and higher settings.

6. Rings (5月28日)

2025DAG06.pdf

We discuss animated rings. The adjoint functor theorem also appears.

7. Modules (6月4日)

2025DAG07.pdf

We discuss animated modules, animated algebras, free/forgetful adjunctions, as well as the symmetric algebra functor, and various compatibilities between these.

8. Schemes (6月11日)

9. Derived categories and stabilisation (6月18日)

10. The cotangent complex (6月25日)

--- 7月2日 No lecture 講義がありません ---

--- 7月9日 No lecture 講義がありません ---

11. Algebraic K-theory (7月16日)