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Wikipedia / 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 is a morphism can be thought of as a universal "linearization" of it, which serves to control the deformation theory. It is constructed as an object in a certain derived category of sheaves 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

are algebraic varieties is a more universal version of the relative Kähler differentials. The most basic motivation for such an object is the exact sequence 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.

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

  • for all other i.

  • is the pullback along i of the inclusion of J of V

  • 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

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 are simplicial rings via the ruleIterating this construction gives a simplicial algebra via the rules.

-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:

which respects the augmentation maps. This map is constructed by choosing a free simplicial C-algebra resolution of D is a free object, the composite hr 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 is a morphism in M induces an exact triangle in the homotopy category,

Cotangent complexes in deformation theory

Setup

and a square-zero infinitesimal, that is a morphism of schemes where the kernelhas the property its square is the zero sheaf, so fitting into cartesian squares of the form then controls the information related to this problem. We can reformulate it as considering the set of extensions of the commutative diagram 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

-schemes as the cone offitting into a distinguished triangle defined similarly. This technique is foundational to Gromov-Witten theory (see below), which studies morphisms from algebraic curves.

Properties of the cotangent complex

Flat base change

Suppose that B and C are A for all . Then there are quasi-isomorphisms

If C is a flat A 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:

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 vanishes for all B-modules M if and only if f is lci. Thus, combined with the above vanishing result we deduce:

  1. The morphism is a perfect complex with Tor amplitude in [-1,0].

has finite projective dimension and B is of finite Tor dimension as an A-module, then f is lci. 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 vanishes for any.

In all of this, it is necessary to suppose that the rings in question are noetherian. For example, let k be a perfect field of characteristic vanishes for any morphism of perfect k-algebras. But not every morphism of perfect k-algebras is lci.

Flat descent

Bhargav Bhatt showed that the cotangent complex satisfies (derived) faithfully flat descent. In other words, for any faithfully flat morphism of R-algebras, one has an equivalence

in the derived category of R, where the right-hand side denotes the homotopy limit of the Čech conerve of f. (The Čech conerve is the cosimplicial object determining the Amitsur complex.) More generally, all the exterior powers of the cotangent complex satisfy faithfully flat descent.

Examples

Smooth schemes

is a finite dimensional affine space 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

.

Local complete intersection

This is given by the complex 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 maps punctures to a fixed target. Since enumerative geometry. Using the distinguished triangleassociated to the composition of morphismsthe cotangent complex can be computed in many situations. In fact, for a complex manifold. From general theory of triangulated categories is quasi-isomorphic to the cone

See also

Notes

References

Applications

  • https://mathoverflow.net/questions/372128/what-is-the-cotangent-complex-good-for

Generalizations

References