Adeel A. Khan

Sheaves on derived spaces

Here are some papers and notes about sheaves on derived spaces/stacks and their cohomology and Borel–Moore homology.

Main papers

Contains most of the derived geometry prerequisites for the below papers. (Joint with Rydh.)

Virtual fundamental classes of derived stacks I

The main paper constructing Borel–Moore homology, cohomology, and fundamental classes of derived stacks. Preliminary draft (revision expected by March 21st).

Fundamental classes in motivic homotopy theory

The formalism that the above paper builds on. Joint with Déglise and Jin.

K-theory and G-theory of derived algebraic stacks

A K-theoretic counterpart to the cohomology and Borel–Moore homology stuff above.

Six operations

Voevodsky’s criterion for constructible categories of coefficients

An improved version of chapter 2 of my thesis. Preliminary draft, some more material to be added later (expected release date: by March 21st).

Perfection in motivic homotopy theory

Topological invariance. Joint with Elmanto.

Motivic homotopy theory

Papers dealing specifically with the stable motivic homotopy category.

Modules over algebraic cobordism

Discusses the role of virtual classes in motivic homotopy theory. Joint with Elmanto, Hoyois, Sosnilo, and Yakerson.

The Morel–Voevodsky localization theorem in spectral algebraic geometry

Subsumes part (chapter 1) of my thesis.

K-theory

Virtual excess intersection theory

Thomason’s excess intersection formula for derived stacks.

Descent by quasi-smooth blow-ups in algebraic K-theory

Projective bundle and blow-up formulas.

Categorical Milnor squares and K-theory of algebraic stacks

Excision for Milnor and blow-up squares. Joint with Bachmann, Ravi, and Sosnilo.

A¹-homotopy invariance in spectral algebraic geometry

Homotopy invariant K-theory and the Bass construction over the sphere spectrum. Joint with Cisinski.

Microlocal theory

Some notes on the Fourier–Sato transform:

1. Overview (2020-01-08)

2. Constructible categories and localization (2020-01-15)

3. The Euler transformation (2020-01-26)

4. A G_m-equivariant sheaf on the affine line (2020-02-01)

5. The homotopy lemma (2020-02-09)

6. Total spaces of perfect complexes (2020-02-12)

7. Homotopy invariance for perfect complexes (2020-02-12)

8. The Fourier–Sato transform (2020-02-16)

9. Involutivity (2020-02-16)

10. The square of ^{^}j_*(1) (2020-02-17)

11. Involutivity: reduction (2020-03-01)

12. Involutivity: connective case (2020-03-01)

13. A functoriality property (2020-03-02)

14. Involutivity: coconnective case (2020-03-13)