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 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 is another morphism, then there is an exact sequence In some sense, therefore, relative Kähler differentials are a right exact functor. 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. 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 ΩX/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. In this case, he defines the cotangent complex of f as an object in the derived category of coherent sheavesX 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 of B by simplicial free A-algebras. Applying the Kähler differential functor to produces a simplicial B-module. The total complex of this simplicial object is the cotangent complexLB/A. The morphism r induces a morphism from the cotangent complex to ΩB/A called the augmentation map. In the homotopy category of simplicial A-algebras, 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 which respects the augmentation maps. This map is constructed by choosing a free simplicial C-algebra resolution of D, say Because is a free object, the composite hr can be lifted to a morphism Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms 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 categoryM. Suppose that is a morphism in M. The cotangent complex is an object in the category of spectra in. A pair of composable morphisms, and induces an exact triangle in the homotopy category,
Suppose that B and C are A-algebras such that for all. Then there are quasi-isomorphisms If C is a flat A-algebra, then the condition that 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.
If A is Noetherian,, and I is generated by a regular sequence, then is a projective module and LB/A is quasi-isomorphic to
Examples
Smooth schemes
Let be smooth. Then the cotangent complex is. In Berthelot's framework, this is clear by taking. In general, étale locally on is a finite dimensional affine space and the morphism is projection, so we may reduce to the situation where and We can take the resolution of 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 be a closed embedding of smooth schemes in. Using the exact triangle corresponding to the morphisms, we may determine the cotangent complex. To do this, note that by the previous example, the cotangent complexes and consist of the Kähler differentials and in the zeroth degree, respectively, and are zero in all other degrees. The exact triangle implies that is nonzero only in the first degree, and in that degree, it is the kernel of the map This kernel is the conormal bundle, and the exact sequence is the conormal exact sequence, so in the first degree, is the conormal bundle.
Local complete intersection
More generally, a local complete intersection morphism with a smooth target has a cotangent complex perfect in amplitude This is given by the complexFor example, the cotangent complex of the twisted cubic in is given by the complex
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 stackswhich are the moduli spaces of mapsfrom genus curves 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, the map, and the target space. Fortunately, all of this deformation theoretic information can be tracked by the cotangent complex. Using the distinguished triangleassociated to the composition of morphismsthe cotangent complex can be computed in many situations. In fact, for a Calabi-Yau manifold its cotangent complex is given by, and a smooth curve, this is given by. From general theory of triangulated categories, the cotangent complex is quasi-isomorphic to the cone
