Derived Algebraic Geometry (DAG)

Derived Algebraic Geometry revolutionizes classical algebraic geometry by incorporating homotopical methods, allowing us to work with "spaces" that remember higher-order information through derived schemes and stacks.

Fundamental Framework

Derived Schemes

A derived scheme is a pair $(X, \mathcal{O}_X)$ where:

  • $X$ is a topological space
  • $\mathcal{O}_X$ is a sheaf of simplicial commutative rings
  • Locally modeled on $\text{Spec}(A)$ for simplicial commutative rings $A$

Toën-Vezzosi Theorem

The ∞-category of derived Artin stacks is the localization:

$$\text{dSt} = \text{sPr}[\text{smooth surjections}^{-1}]$$

This provides a universal solution to moduli problems in the derived setting.

Derived Stack Structure

Moduli Problems in DAG

Classical Moduli Space

Singular points where classical theory breaks down

Derived Enhancement

Smooth derived structure resolving singularities

The Cotangent Complex

For a morphism $f: X \to Y$ of derived schemes, the cotangent complex $\mathbb{L}_{X/Y}$ is the fundamental deformation-theoretic invariant, generalizing Kähler differentials.

Example: Intersection Theory

For schemes $X, Y \hookrightarrow Z$, the derived intersection $X \times^h_Z Y$ has cotangent complex:

$$\mathbb{L}_{X \times^h_Z Y} \simeq \mathbb{L}_X \oplus \mathbb{L}_Y \to \mathbb{L}_Z$$

The failure of this to be an isomorphism measures the "excess intersection".

Derived Categories and Triangulated Structure

// Derived category construction in pseudocode DerivedCategory(A) { Objects: Chain complexes of A-modules Morphisms: Chain maps up to homotopy Triangulated structure: - Distinguished triangles: X → Y → Z → X[1] - Rotation axiom - Octahedral axiom return LocalizeAt(ChainComplexes(A), QuasiIsomorphisms) }

Spectral Sequences in DAG

The spectral sequence of a filtered complex provides computational tools for derived invariants.

Revolutionary Applications

Virtual Fundamental Classes

In Gromov-Witten theory, derived structures provide well-defined integration over moduli spaces that are singular in the classical sense.

Derived Deformation Theory

The cotangent complex controls deformations, obstructions live in $\text{Ext}^2$, and higher $\text{Ext}$ groups control higher-order phenomena.

Shifted Symplectic Structures

Pantev-Toën-Vaquié-Vezzosi: Many moduli spaces carry n-shifted symplectic structures, explaining mysterious dualities.

Homotopy Limits and Colimits

Derived fiber products and pushouts encode higher-order intersection data.

Foundational Papers

HAG I & II: Homotopical Algebraic Geometry
Bertrand Toën, Gabriele Vezzosi (2008)
arXiv:math/0404373

The foundational papers establishing derived algebraic geometry.

Derived Algebraic Geometry
Jacob Lurie (2009-2018)
DAG Series

Comprehensive development using ∞-categorical methods.

Shifted Symplectic Structures
Pantev, Toën, Vaquié, Vezzosi (2013)
arXiv:1111.3209

Revolutionary discovery of shifted symplectic geometry on moduli spaces.

Derived Categories and Their Applications
Maxim Kontsevich (Various)

Homological mirror symmetry and derived categories of coherent sheaves.

Leading Research Centers

  • MIT - Jacob Lurie's systematic development
  • Princeton/IAS - Higher algebra and topology
  • Stanford - Applications to algebraic topology
  • Toulouse (IMT) - Bertrand Toën's group
  • Paris (IHES) - Maxim Kontsevich
  • Imperial College London - Derived categories group