Derived Algebraic GeometryEdit
Derived Algebraic Geometry is a modern framework that extends classical algebraic geometry by encoding derived or higher-homotopical information directly into geometric objects. By enriching the algebra of functions with homotopical data, it provides a natural language for handling subtle issues that arise in intersections, deformations, and moduli problems. The subject draws on ideas from homotopy theory, higher category theory, and algebraic geometry, and it has developed into several closely related formalisms, each with its own advantages for the problems at hand. Central goals include recovering higher-order obstruction information, organizing deformation theory, and providing robust foundations for moduli spaces that carry intrinsic virtual or derived structure. See algebraic geometry and cotangent complex for foundational notions that enter the derived setting.
In broad terms, derived algebraic geometry replaces classical commutative algebras of functions with homotopically enhanced versions, such as simplicial commutative algebras or E_infty-algebras, and studies spaces that locally look like spectra of such algebras. This yields derived schemes, derived stacks, and, in broader families, spectral algebraic geometry as a parallel language. The development has been shaped by two complementary traditions: one based on model-categorical and homotopical algebra in algebraic geometry, and another based on ∞-categories and higher topos theory, culminating in a highly flexible and conceptually unified picture. Foundational strands run from early work on the cotangent complex and deformation theory to the modern ∞-categorical formulations.
History and foundations
The impetus for a derived viewpoint goes back to classical deformation theory and intersection theory. The cotangent complex, introduced by Illusie, provides a derived-mechanical account of infinitesimal deformations and obstructions, encoding how a geometric object can be deformed inside a family. This perspective makes it clear that many geometric questions are controlled not just by ordinary tangent spaces but by derived data that capture higher-order deformations. See cotangent complex.
Two influential early programs established the derived paradigm in algebraic geometry. The first, developed by Toën and Vezzosi, systematized the use of homotopical methods in algebraic geometry via model categories and simplicial commutative algebras, giving rise to derived algebraic geometry as a robust framework for doing geometry with higher homotopical information. The second stream, advanced by Jacob Lurie and collaborators, casts the theory in the language of ∞-categories and higher topos theory, culminating in a sequence of works often summarized as DAG I–IV, with later developments expanding into related areas such as spectral algebraic geometry and more.
Derived algebraic geometry also connects to classical algebraic geometry through the idea of derived enhancements of familiar objects. A central construction is the derived fiber product, or homotopy pullback, which replaces the ordinary fiber product when intersections are not transverse. This mechanism yields refined intersection theories and more faithful deformation-theoretic information.
Key foundational concepts associated with these foundations include: - The cotangent complex as the universal object governing deformations and obstructions. - Derived schemes, locally modeled on simplicial commutative algebras or E_infty-algebras. - Derived stacks, which generalize sheaves of spaces on a site of derived affine schemes and underpin modern moduli theory. - The idea of working in an ∞-categorical setting to manage higher morphisms and homotopies in a coherent way.
Core constructions and objects
Derived schemes: These are geometric objects locally described by spectra of homotopically enhanced algebras, such as simplicial commutative algebras or E_infty-algebras. They keep track of higher-order information that classical schemes may miss, particularly in the presence of nontransverse intersections.
Derived stacks: Extending the notion of stacks, derived stacks live on a site of derived affine schemes and are sheaves of spaces (or ∞-groupoids) that respect descent. They provide a natural setting for refined moduli problems and virtual techniques.
Derived fiber products: When taking intersections in algebraic geometry, the derived fiber product retains higher-order information about how components meet. This leads to more accurate invariants and obstruction theories, notably in moduli problems.
∞-categorical foundations: The use of ∞-categories allows a principled treatment of higher morphisms and homotopies, which are intrinsic to derived geometric objects. This viewpoint is central to many modern developments in the field.
Spectral algebraic geometry: A parallel strand, often associated with the perspective of E_infty-ring spectra, yields spectral schemes and related objects. This approach emphasizes a homotopical refinement of the algebraic-geometry toolkit.
Derived deformations and obstruction theory: By encoding deformation problems in the derived setting, one obtains a more complete and stable understanding of how objects vary in families, including obstructions and higher obstructions.
Techniques, applications, and examples
Intersection theory: Derived intersections correct multiplicities in situations where classical intersections are singular or nontransverse. This provides a more faithful account of the geometry and its invariants.
Moduli problems: Moduli spaces often carry naturally derived structures that reflect infinitesimal and obstruction information. Derived enhancements of moduli stacks give more robust foundations for counting problems, virtual fundamental classes, and localization techniques.
Deformation theory: The cotangent complex governs deformations, with derived methods enabling systematic obstruction theories. This is particularly important for questions about smoothness, rigidity, and versal deformations.
Computational and conceptual clarity: While the machinery can be heavy, the derived viewpoint often clarifies why certain phenomena occur and provides unifying explanations across diverse problems, from classical examples to modern moduli questions.
Connections to physics: In some lines of mathematical physics and string theory, derived geometry provides a natural language for describing extended objects, branes, and associated categories. The formalism helps organize how physical theories encode geometric and topological data.
Controversies and debates
As with any highly abstract and transformative framework, derived algebraic geometry has sparked discussions about balance between abstraction and computability. Major themes include: - Access and practicality: Critics sometimes point to the substantial technical overhead required to work in ∞-categorical or model-categorical settings, arguing that this can hinder concrete computations or concrete geometric intuition. Proponents respond that the derived framework resolves fundamental obstructions and provides a coherent global picture that simpler approaches miss. - Foundations and formalism: Different communities favor different foundational choices (model categories vs ∞-categories, for instance). Each has its advantages and trade-offs in terms of rigor, flexibility, and interoperability with other branches of mathematics. - Scope and necessity: Some practitioners debate how essential derived enhancements are for particular problems. In many situations, classical methods suffice, while in others the derived perspective yields essential insights or corrects classical shortcuts. - Interdisciplinary adoption: The extent to which derived methods should influence broader areas of algebraic geometry, number theory, or mathematical physics is a topic of ongoing discussion. Advocates emphasize long-term unification and conceptual clarity, while skeptics emphasize practical tractability.
Relationships to other areas
Traditional algebraic geometry: DAG builds on classical objects such as schemes, sheaves, and moduli spaces, but enriches them with higher-homotopical data to handle phenomena that classical methods gloss over.
Deformation theory and obstruction theory: The derived framework integrates deformation theory deeply, with the cotangent complex and derived tangent data playing central roles.
Higher category theory and topos theory: The ∞-categorical language is not merely a stylistic choice; it provides the natural setting to handle higher morphisms and descent in a coherent way.
Moduli theory and virtual techniques: Derived structures underpin modern approaches to moduli problems, including refined invariants and virtual fundamental classes used in enumerative geometry and related fields.
Connections to physics and representation theory: Derived geometry intersects with topics in mathematical physics and the study of categories of branes, as well as with higher-categorical approaches in representation theory.