Loterre: Mathematics: cotangent complex

geometry > algebraic geometry > cotangent complex

topology > algebraic topology > homotopical algebra > cotangent complex

... > abstract algebra > algebraic structure > homological algebra > homotopical algebra > cotangent complex

topology > algebraic topology > homotopy theory > cotangent complex

category theory > cotangent complex

cotangent complex

In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If ${\\displaystyle f:X\ o Y}$ is a morphism of geometric or algebraic objects, the corresponding cotangent complex ${\\displaystyle \\mathbf {L} _{X/Y}^{\ullet }}$ can be thought of as a universal "linearization" of it, which serves to control the deformation theory of ${\\displaystyle f}$ . It is constructed as an object in a certain derived category of sheaves on ${\\displaystyle X}$ using the methods of homotopical algebra.
Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this definition to the general situation of a morphism of ringed topoi, thereby incorporating morphisms of ringed spaces, schemes, and algebraic spaces into the theory.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cotangent_complex)

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