Derived algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide a local chart, are replaced by ring spectra in algebraic topology, whose higher homotopy accounts 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, and provides a natural setting for cotangent complexes and such in deformation theory (cf. F. Francis).
Basic objects of study in the field are derived schemes and derived stacks; they generalize, for instance, differential graded schemes. The oft-cited example is Serre's intersection formula. In the usual formulation, the formula involves 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 just a theory; the derived intersection theory has yet to be developed.)
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