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 ${\displaystyle f:X\to Y}$ is a morphism of geometric or algebraic objects, the corresponding cotangent complex ${\displaystyle \mathbf {L} _{X/Y}^{\bullet }}$ 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.

Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are algebraic varieties and that ${\displaystyle f:X\to Y}$ is a morphism between them. The cotangent complex of ${\displaystyle f}$ is a more universal version of the relative Kähler differentials ${\displaystyle \Omega _{X/Y}}$. The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If ${\displaystyle Z}$ is another variety, and if ${\displaystyle g:Y\to Z}$ 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 ${\displaystyle T^{i}}$ and imperfection modules. Most of these were motivated by deformation theory.

This sequence is exact on the left if the morphism ${\displaystyle f}$ 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.

Cotangent complexes appeared in multiple and partially incompatible versions of increasing generality in the early 1960s. The first instance of the related homology functors in the restricted context of field extensions appeared in Cartier (1956). Alexander Grothendieck then developed an early version of cotangent complexes in 1961 for his general Riemann-Roch theorem in algebraic geometry in order to have a theory of virtual tangent bundles. This is the version described by Pierre Berthelot in SGA 6, Exposé VIII. It only applies when f is a smoothable morphism (one that factors into a closed immersion followed by a smooth morphism). In this case, the cotangent complex of f as an object in the derived category of coherent sheaves on X is given as follows: