: Mathématiques: derived algebraic geometry

geometry > algebraic geometry > derived algebraic geometry

topology > algebraic topology > homotopical algebra > derived algebraic geometry

... > abstract algebra > algebraic structure > homological algebra > homotopical algebra > derived algebraic geometry

algebra > abstract algebra > algebraic structure > ring theory > derived algebraic geometry

derived algebraic geometry

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 ${\\displaystyle \\mathbb {Q} }$ ), simplicial commutative rings or ${\\displaystyle E_{\\infty }}$ -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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Derived_algebraic_geometry)

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