Derived algebraic geometry. (English) Zbl 1314.14005
Derived algebraic geometry is a relatively recent branch of mathematics that extends the framework of usual algebraic geometry with the purpose to tackle geometrically special situations effectively. While in classical algebraic geometry the elementary building blocks are spectra of commutative rings, the viewpoint of derived algebraic geometry is based on ring spectra occurring in algebraic topology and homotopical algebra, that, is, on certain non-commutative rings. The mathematical foundations of derived algebraic geometry where mostly developed in the course of the last fifteen years, especially in the works of J. Lurie, B. Toën, G. Vezzosi, and others. Today the subject has a rather large spectrum of interactions with other mathematical theories, including moduli theory, arithmetic geometry, geometric representation theory, and mathematical physics. For example, some recent progress in the study of the geometric Langlands correspondence, topological modular forms, deformation quantization, or \(p\)-adic Hodge theory was achieved by applying the new methods of derived algebraic geometry.
In the survey article under review, the basic concepts and results in derived algebraic geometry are comprehensively described, together with related historical explanations and an illustration of some recent applications in deformation quantization.
More precisely, Section 1 presents some selected historical developments of the various ideas that finally led to the topic of derived algebraic geometry. Section 2 introduces the language of the subject via higher category theory and introduces the notion of derived schemes. Section 3 discusses some characteristic properties of derived schemes as well as derived moduli problems, derived Artin stacks, and other contexts of derived algebraic geometry. Section 4 presents the formal geometry of derived schemes and derived stacks, mainly with a view toward cotangent complexes and obstruction theory, formal descent, tangent differential graded Lie algebras, derived loop spaces and algebraic de Rham theory. The final Section 5 deals with symplectic, Poisson and Lagrangian structures in the derived setting. This discussion includes the notion of differential forms on derived schemes and on derived stacks, shifted symplectic and Lagrangian structures as well as the notion of polyvectors and shifted Poisson structures. A number of illustrating examples supplements the presentation of this highly topical material of current research.
The rich bibliography refers to 146 related research papers for further, more detailed reading about the wealth of ideas, concepts, techniques, and deep results outlined in this highly instructive survey article.
In the survey article under review, the basic concepts and results in derived algebraic geometry are comprehensively described, together with related historical explanations and an illustration of some recent applications in deformation quantization.
More precisely, Section 1 presents some selected historical developments of the various ideas that finally led to the topic of derived algebraic geometry. Section 2 introduces the language of the subject via higher category theory and introduces the notion of derived schemes. Section 3 discusses some characteristic properties of derived schemes as well as derived moduli problems, derived Artin stacks, and other contexts of derived algebraic geometry. Section 4 presents the formal geometry of derived schemes and derived stacks, mainly with a view toward cotangent complexes and obstruction theory, formal descent, tangent differential graded Lie algebras, derived loop spaces and algebraic de Rham theory. The final Section 5 deals with symplectic, Poisson and Lagrangian structures in the derived setting. This discussion includes the notion of differential forms on derived schemes and on derived stacks, shifted symplectic and Lagrangian structures as well as the notion of polyvectors and shifted Poisson structures. A number of illustrating examples supplements the presentation of this highly topical material of current research.
The rich bibliography refers to 146 related research papers for further, more detailed reading about the wealth of ideas, concepts, techniques, and deep results outlined in this highly instructive survey article.
Reviewer: Werner Kleinert (Berlin)
MSC:
| 14A20 | Generalizations (algebraic spaces, stacks) |
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
| 18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 53D50 | Geometric quantization |
Keywords:
derived algebraic geometry; derived scheme; derived moduli; derived stack; cotangent complex; formal geometry; \(\infty\)-categoriesReferences:
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