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 .[1][2] 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

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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.[3] 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

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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.[4] It only applies when f is a smoothable morphism (one that factors into a closed immersion followed by a smooth morphism).[5] 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,[6] and for a smoothable complete intersection morphism, this complex is perfect.[7] Furthermore, if g : YZ 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[8] and Lichtenbaum and Schlessinger.[9] Their theories extended to cotangent complexes of length 3, thus capturing more information.

The definition of the cotangent complex

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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 rule

 

Iterating this construction gives a simplicial algebra

 

where the horizontal maps come from composing the augmentation maps for the various choices. For example, there are two augmentation maps   via the rules

 

which 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

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Setup

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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 kernel

 

has 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 form

 

A 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 diagram

 

which is a homological problem. Then, the set of such diagrams whose kernel is   is isomorphic to the abelian group

 

showing the cotangent complex controls the set of deformations available.[1] Furthermore, from the other direction, if there is a short exact sequence

 

there exists a corresponding element

 

whose vanishing implies it is a solution to the deformation problem given above. Furthermore, the group

 

controls the set of automorphisms for any fixed solution to the deformation problem.

Some important implications

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One of the most geometrically important properties of the cotangent complex is the fact that given a morphism of  -schemes

 

we can form the relative cotangent complex   as the cone of

 

fitting into a distinguished triangle

 

This 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 is a powerful technique and 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

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Flat base change

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Suppose that B and C are A-algebras such that   for all q > 0. Then there are quasi-isomorphisms[10]

 

If C is a flat A-algebra, then the condition that   vanishes for q > 0 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

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Let f : AB. Then:[11][12]

  • 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].[13]
  • 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 p > 0, then  .[14]

Characterization of local complete intersections

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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 f : AB 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.[15] Thus, combined with the above vanishing result we deduce:

The morphism f : AB 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.[16] This was proven by Luchezar Avramov in a 1999 Annals paper.[17] 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  .[18]

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 p > 0. Then as noted above,   vanishes for any morphism AB of perfect k-algebras. But not every morphism of perfect k-algebras is lci.[19]

Flat descent

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Bhargav Bhatt showed that the cotangent complex satisfies (derived) faithfully flat descent.[20] In other words, for any faithfully flat morphism f : AB 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   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

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Smooth schemes

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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

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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

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More generally, a local complete intersection morphism   with a smooth target has a cotangent complex perfect in amplitude   This is given by the complex

 

For example, the cotangent complex of the twisted cubic   in   is given by the complex

 

Cotangent complexes in Gromov-Witten theory

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In Gromov–Witten theory mathematicians study the enumerative geometric invariants of n-pointed curves on spaces. In general, there are algebraic stacks

 

which are the moduli spaces of maps

 

from genus   curves with   punctures 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 triangle

 

associated to the composition of morphisms

 

the cotangent complex can be computed in many situations. In fact, for a complex manifold  , its cotangent complex is given by  , and a smooth  -punctured curve  , this is given by  . From general theory of triangulated categories, the cotangent complex   is quasi-isomorphic to the cone

 

See also

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Notes

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  1. ^ a b "Section 91.21 (08UX): Deformations of ringed spaces and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.
  2. ^ "Section 91.23 (08V3): Deformations of ringed topoi and the cotangent complex—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-12-02.
  3. ^ Grothendieck 1967, Proposition 17.2.5
  4. ^ Berthelot 1966, VIII Proposition 2.2
  5. ^ (Grothendieck 1968, p. 4)
  6. ^ Berthelot 1966, VIII Proposition 2.2
  7. ^ Berthelot 1966, VIII Proposition 2.4
  8. ^ (Gerstenhaber 1964)
  9. ^ (Lichenbaum; Schlessinger 1967)
  10. ^ Quillen 1970, Theorem 5.3
  11. ^ Quillen 1970, Theorem 5.4
  12. ^ Quillen 1970, Corollary 6.14
  13. ^ "Section 91.14 (08SH): The cotangent complex of a local complete intersection—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-21.
  14. ^ Mathew, Akhil (2022-03-02). "Some recent advances in topological Hochschild homology". Bull. London Math. Soc. 54 (1). Prop. 3.5. arXiv:2101.00668. doi:10.1112/blms.12558. S2CID 230435604.
  15. ^ Lichtenbaum–Schlessinger 1967, Corollary 3.2.2.
  16. ^ Quillen 1970, Conjecture 5.7.
  17. ^ Avramov, Luchezar L. (1999). "Locally Complete Intersection Homomorphisms and a Conjecture of Quillen on the Vanishing of Cotangent Homology". Annals of Mathematics. 150 (2): 455–487. arXiv:math/9909192. doi:10.2307/121087. ISSN 0003-486X. JSTOR 121087. S2CID 17250847.
  18. ^ Briggs, Benjamin; Iyengar, Srikanth (2022). "Rigidity properties of the cotangent complex". Journal of the American Mathematical Society. 36: 291–310. arXiv:2010.13314. doi:10.1090/jams/1000. ISSN 0894-0347. S2CID 225070623.
  19. ^ Haine, Peter (2020-04-02). "The lci locus of the Hilbert scheme of points & the cotangent complex" (PDF). p. 11. Archived (PDF) from the original on 2021-07-08.
  20. ^ Bhatt, Bhargav; Morrow, Matthew; Scholze, Peter (2019-06-01). "Topological Hochschild homology and integral p-adic Hodge theory". Publications mathématiques de l'IHÉS. 129 (1): 199–310. doi:10.1007/s10240-019-00106-9. ISSN 1618-1913. S2CID 254165606.

References

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Applications

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Generalizations

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References

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