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.
Motivation.
Suppose that "X" and "Y" are algebraic varieties and that is a morphism between them. The cotangent complex of "f" is a more universal version of the relative Kähler differentials Ω"X"/"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 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 "T""i" 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.
Another natural exact sequence related to Kähler differentials is the conormal exact sequence. If "f" is a closed immersion with ideal sheaf "I", then there is an exact sequence
This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of "f", and the relative differentials Ω"X"/"Y" have vanished because a closed immersion is formally unramified. If "f" is the inclusion of a smooth subvariety, then this sequence is a short exact sequence. This suggests that the cotangent complex of the inclusion of a smooth variety is equivalent to the conormal sheaf shifted by one term.
Early work on cotangent complexes.
The cotangent complex dates back at least to SGA 6 VIII 2, where Pierre Berthelot gave a definition when "f" is a "smoothable" morphism, meaning there is a scheme "V" and morphisms and such that , "i" is a closed immersion, and "h" is a smooth morphism. (For example, all projective morphisms are smoothable, since "V" can be taken to be a projective bundle over "Y".) In this case, he defines the cotangent complex of "f" as an object in the derived category of coherent sheaves "X" as follows:
Berthelot proves that this definition is independent of the choice of "V" and that for a smoothable complete intersection morphism, this complex is perfect. Furthermore, he proves that if is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an exact triangle
The definition of the cotangent complex.
The correct definition of the cotangent complex begins in the homotopical setting. Quillen and André worked with the simplicial commutative rings, while Illusie worked with simplicial ringed topoi. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that "A" and "B" are simplicial rings and that "B" is an "A"-algebra. Choose a resolution formula_10 of "B" by simplicial free "A"-algebras. Applying the Kähler differential functor to formula_11 produces a simplicial "B"-module. The total complex of this simplicial object is the cotangent complex "L""B"/"A". The morphism "r" induces a morphism from the cotangent complex to Ω"B"/"A" called the augmentation map. In the homotopy category of simplicial "A"-algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler differential functor.
Given a commutative square as follows:
there is a morphism of cotangent complexes formula_12 which respects the augmentation maps. This map is constructed by choosing a free simplicial "C"-algebra resolution of "D", say formula_13 Because formula_11 is a free object, the composite "hr" can be lifted to a morphism formula_15 Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms formula_16 this produces the sequence
There is a connecting homomorphism, 
which turns this sequence into an exact triangle.
The cotangent complex can also be defined in any combinatorial model category "M". Suppose that formula_19 is a morphism in "M". The cotangent complex formula_20 (or formula_21) is an object in the category of spectra in formula_22. A pair of composable morphisms, formula_19 and formula_24 induces an exact triangle in the homotopy category, 
Properties of the cotangent complex.
Flat base change.
Suppose that "B" and "C" are "A"-algebras such that formula_26 for all . Then there are quasi-isomorphisms
If "C" is a flat "A"-algebra, then the condition that formula_28 vanishes for is automatic. The first formula then proves that the construction of the cotangent complex is local on the base in the flat topology.
Vanishing properties.
Let . Then:
Examples.
Smooth schemes.
Let formula_31 be smooth. Then the cotangent complex is formula_32. In Berthelot's framework, this is clear by taking formula_33. In general, étale locally on formula_34 is a finite dimensional affine space and the morphism formula_35 is projection, so we may reduce to the situation where formula_36 and formula_37 We can take the resolution of formula_38 to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.
Closed embeddings in smooth schemes.
Let formula_39 be a closed embedding of smooth schemes in formula_40. Using the exact triangle corresponding to the morphisms formula_41, we may determine the cotangent complex formula_42. To do this, note that by the previous example, the cotangent complexes formula_43 and formula_44 consist of the Kähler differentials formula_32 and formula_46 in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that formula_42 is nonzero only in the first degree, and in that degree, it is the kernel of the map formula_48 This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, formula_42 is the conormal bundle formula_50.
Local complete intersection.
More generally, a local complete intersection morphism formula_51 with a smooth target has a cotangent complex perfect in amplitude formula_52 This is given by the complexformula_53For example, the cotangent complex of the twisted cubic formula_54 in formula_55 is given by the complexformula_56
Cotangent complexes in Gromov-Witten theory.
In Gromov–Witten theory mathematicians study the enumerative geometric invariants of n-pointed curves on spaces. In general, there are algebraic stacksformula_57which are the moduli spaces of mapsformula_58from genus formula_59 curves with formula_60 punctures to a fixed target. Since enumerative geometry studies the generic behavior of such maps, the deformation theory controlling these kinds of problems requires the deformation of the curve formula_61, the map formula_62, and the target space formula_54. Fortunately, all of this deformation theoretic information can be tracked by the cotangent complex formula_64. Using the distinguished triangleformula_65associated to the composition of morphismsformula_66the cotangent complex can be computed in many situations. In fact, for a complex manifold formula_54, its cotangent complex is given by formula_68, and a smooth formula_60-punctured curve formula_61, this is given by formula_71. From general theory of triangulated categories, the cotangent complex formula_64 is quasi-isomorphic to the coneformula_73