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Concept
Derived algebraic geometry
Résumé
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications.
Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not yield the correct intersection number. In the derived context, one takes the derived tensor product , whose higher homotopy is higher Tor, whose Spec is not a scheme but a derived scheme. Hence, the "derived" fiber product yields the correct intersection number. (Currently this is hypothetical; the derived intersection theory has yet to be developed.)
The term "derived" is used in the same way as derived functor or , in the sense that the category of commutative rings is being replaced with a of "derived rings." In classical algebraic geometry, the derived category of quasi-coherent sheaves is viewed as a , but it has natural enhancement to a , which can be thought of as the analogue of an .
Derived algebraic geometry is fundamentally the study of geometric objects using homological algebra and homotopy. Since objects in this field should encode the homological and homotopy information, there are various notions of what derived spaces encapsulate.
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