Cotangent complex
In mathematics the cotangent complex is roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M. André independently came up with a definition that works in all cases.
Suppose that X and Y are algebraic varieties and that f : X → Y is a morphism between them. The cotangent complex of f is a more universal version of the relative Kähler differentials ΩX/Y. The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If Z is another variety, and if g : Y → Z is another morphism, then there is an exact sequence
In some sense, therefore, relative Kähler differentials are a right exact functor. (Literally this is not true, however, because the category of algebraic varieties is not an abelian category, and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the Lichtenbaum–Schlessinger functors Ti and imperfection modules. Most of these were motivated by deformation theory.
This sequence is exact on the left if the morphism f is smooth. If Ω admitted a first derived functor, then exactness on the left would imply that the connecting homomorphism vanished, and this would certainly be true if the first derived functor of f, whatever it was, vanished. Therefore a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials.
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