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 is a morphism of geometric or algebraic objects, the corresponding cotangent complex can be thought of as a universal "linearization" of it, which serves to control the deformation theory of . It is constructed as an object in a certain derived category of sheaves on 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.
Motivation Suppose that and are algebraic varieties and that is a morphism between them. The cotangent complex of is a more universal version of the relative Kähler differentials . The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If 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 and imperfection modules. Most of these were motivated by deformation theory.
This sequence is exact on the left if the morphism 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 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:
If J is the ideal of X in V, then for all other i. The differential is the pullback along i of the inclusion of J in the structure sheaf of V followed by the universal derivation All other differentials are zero. This definition is independent of the choice of V, and for a smoothable complete intersection morphism, this complex is perfect. Furthermore, if 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). However, like the theory of 1961, this produced a cotangent complex of length 2 only, corresponding to the truncation of the full complex which was not yet known at the time. This approach was published later in Grothendieck (1968). 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 and Lichtenbaum and Schlessinger. Their theories extended to cotangent complexes of length 3, thus capturing more information.
The definition of the cotangent complex The correct definition of the cotangent complex begins in the homotopical setting. 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. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that and are simplicial rings and that is an -algebra. Choose a resolution of by simplicial free -algebras. Such a resolution of can be constructed by using the free commutative -algebra functor which takes a set and yields the free -algebra . For an -algebra , this comes with a natural augmentation map which maps a formal sum of elements of to an element of via the ruleIterating this construction gives a simplicial algebrawhere the horizontal maps come from composing the augmentation maps for the various choices. For example, there are two augmentation maps via the ruleswhich can be adapted to each of the free -algebras .
Applying the Kähler differential functor to produces a simplicial -module. The total complex of this simplicial object is the cotangent complex LB/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 (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 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 category M. Suppose that is a morphism in M. The cotangent complex (or ) is an object in the category of spectra in . A pair of composable morphisms, and induces an exact triangle in the homotopy category,
Cotangent complexes in deformation theory
Setup One of the first direct applications of the cotangent complex is in deformation theory. For example, if we have a scheme and a square-zero infinitesimal thickening , that is a morphism of schemes where the kernelhas the property its square is the zero sheaf, so one of the fundamental questions in deformation theory is to construct the set of fitting into cartesian squares of the formA couple examples to keep in mind is extending schemes defined over to , or schemes defined over a field of characteristic to the ring where . The cotangent complex then controls the information related to this problem. We can reformulate it as considering the set of extensions of the commutative diagramwhich is a homological problem. Then, the set of such diagrams whose kernel is is isomorphic to the abelian groupshowing the cotangent complex controls the set of deformations available. Furthermore, from the other direction, if there is a short exact sequencethere exists a corresponding elementwhose vanishing implies it is a solution to the deformation problem given above. Furthermore, the groupcontrols the set of automorphisms for any fixed solution to the deformation problem.
Some important implications One of the most geometrically important properties of the cotangent complex is the fact that given a morphism of -schemeswe can form the relative cotangent complex as the cone offitting into a distinguished triangleThis is one of the pillars for cotangent complexes because it implies the deformations of the morphism of -schemes is controlled by this complex. In particular, controls deformations of as a fixed morphism in , deformations of which can extend , meaning there is a morphism which factors through the projection map composed with , and deformations of defined similarly. This technique is foundational to Gromov-Witten theory (see below), which studies morphisms from algebraic curves of a fixed genus and fixed number of punctures to a scheme .
Properties of the cotangent complex
Flat base change 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.
Vanishing properties Let . Then:
If B is a localization of A, then . If f is an étale morphism, then . If f is a smooth morphism, then is quasi-isomorphic to . In particular, it has projective dimension zero. If f is a local complete intersection morphism, then is a perfect complex with Tor amplitude in [-1,0]. If A is Noetherian, , and is generated by a regular sequence, then is a projective module and is quasi-isomorphic to If f is a morphism of perfect k-algebras over a perfect field k of characteristic , then .
Characterization of local complete intersections The theory of the cotangent complex allows one to give a homological characterization of local complete intersection (lci) morphisms, at least under noetherian assumptions. Let be a morphism of noetherian rings such that B is a finitely generated A-algebra. As reinterpreted by Quillen, work of Lichtenbaum–Schlessinger shows that the second André–Quillen homology group vanishes for all B-modules M if and only if f is lci.Lichtenbaum–Schlessinger 1967, Corollary 3.2.2. Thus, combined with the above vanishing result we deduce:
The morphism is lci if and only if is a perfect complex with Tor amplitude in [-1,0].
Quillen further conjectured that if the cotangent complex has finite projective dimension and B is of finite Tor dimension as an A-module, then f is lci.Quillen 1970, Conjecture 5.7. This was proven by Luchezar Avramov in a 1999 Annals paper. Avramov also extended the notion of lci morphism to the non-finite type setting, assuming only that the morphism f is locally of finite flat dimension, and he proved that the same homological characterization of lci morphisms holds there (apart from no longer being perfect). Avramov's result was recently improved by Briggs–Iyengar, who showed that the lci property follows once one establishes that vanishes for any single .