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.
can be thought of as a universal "linearization" of it, which serves to control the deformation theory of
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.
The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms.
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.
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.
[3] 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.
[7] Furthermore, if g : Y → Z is another smoothable complete intersection morphism and if an additional technical condition is satisfied, then there is an exact triangle In 1963 Grothendieck developed a more general construction that removes the restriction to smoothable morphisms (which also works in contexts other than algebraic geometry).
At the same time in the early 1960s, largely similar theories were independently introduced for commutative rings (corresponding to the "local" case of affine schemes in algebraic geometry) by Gerstenhaber[8] and Lichtenbaum and Schlessinger.
[9] Their theories extended to cotangent complexes of length 3, thus capturing more information.
Quillen and André worked with simplicial commutative rings, while Illusie worked more generally with simplicial ringed topoi, thus covering "global" theory on various types of geometric spaces.
This map is constructed by choosing a free simplicial C-algebra resolution of D, say
controls the set of automorphisms for any fixed solution to the deformation problem.
One of the most geometrically important properties of the cotangent complex is the fact that given a morphism of
The first formula then proves that the construction of the cotangent complex is local on the base in the flat topology.
Then:[11][12] The theory of the cotangent complex allows one to give a homological characterization of local complete intersection (lci) morphisms, at least under noetherian assumptions.
[15] Thus, combined with the above vanishing result we deduce: Quillen further conjectured that if the cotangent complex
Avramov's result was recently improved by Briggs–Iyengar, who showed that the lci property follows once one establishes that
[19] Bhargav Bhatt showed that the cotangent complex satisfies (derived) faithfully flat descent.
[20] In other words, for any faithfully flat morphism f : A → B of R-algebras, one has an equivalence in the derived category of R, where the right-hand side denotes the homotopy limit of the cosimplicial object given by taking
More generally, all the exterior powers of the cotangent complex satisfy faithfully flat descent.
to be the identity map, and then it is clear that the cotangent complex is the same as the Kähler differentials.
In Gromov–Witten theory mathematicians study the enumerative geometric invariants of n-pointed curves on spaces.
Fortunately, all of this deformation theoretic information can be tracked by the cotangent complex