nLab
cotangent complex

Contents

Idea

The notion of cotangent complex is a derived or (∞,1)-categorical refinement of the notion of Kähler differentials.

Traditionally this has been conceived in terms of model category presentations. This we discuss in the section

From the nPOV, the notion of cotangent complex has a more intrinsic description as being the left adjoint to the tangent (∞,1)-category projection. This we discuss in the section

Motivation

The construction of moduli spaces depends strongly on local properties like smoothness and transversality of intersections when trying to represent the functor of assigning families of objects to the varying base of family. When passing to the classes of equivalent objects, one faces the problem of having nontrivial automorphisms.

At the infinitesimal level automorphisms correspond to the derivations. Taking derivations is represented by the module of relative Kähler differentials which sufficies in good cases. Its correct derived replacement is the cotangent complex of Grothendieck-Illusie. One can typically split the information about a map of higher rings into its “discrete part” and infinitesimal obstruction theory governed by the cotangent complex.

Model category presentation

Quillen’s definition: left derived functor of Kähler differentials

The cotangent complex functor is effectively the left derived functor of the Kähler differentials assignment.

To talk about the nonabelian derived functors, Quillen introduced a model structure on the category of simplicial commutative rings.

Given a morphism f:ABf: A\to B in CRing, which makes BB an AA-algebra, the fiber of the tangent category AbGr(AAlg/B)AbGr(A Alg/B) of abelian group objects in the slice category AAlg/BA Alg/B of AA-algebras over BB is equivalent both to the category of BB-modules and the trivial (= square zero) extensions of AA by BB-modules.

In particular we can consider the forgetful functor AbGr(AAlg/B)AAlg/BAbGr(A Alg/B)\to A Alg/B which has a left adjoint Ω A/BAb B/A:AAlg/BAbGr(AAlg/B) BMod\Omega_{A/B} Ab_{B/A} : A Alg/B\to AbGr(A Alg/B)\cong {}_B Mod. This is the Kähler differentials functor.

All said is true for simplicial commutative rings as well.

Definition

The relative cotangent complex functor is the left derived functor

𝕃Ω B/A:sAAlg/BsAbGr(AAlg/B)sBMod \mathbb{L} \Omega_{B/A} : s A Alg/B\to s AbGr(A Alg/B)\cong s B Mod

Its value on BB is the relative cotangent complex L B/AL_{B/A}

The Andre-Quillen cohomology of RR is the cohomology of 𝕃Ω(R)\mathbb{L}\Omega(R).

Explicit resolutions

Here is one way to compute the required cofibrant resolution for the construction of the left derived functor for the case that A=kA = k is a field.

Let P:CAlgCAlgP : CAlg \to CAlg be the comonad induced by the adjunction

U:CAlgSet:k[] U : CAlg \to Set : k[-]

that sends a commutative kk-algebra RR to the polynomial algebra on its underlying set.

Let P RP_\bullet R be the corresponding bar construction simplicial algebra. The canonical morphism P RRP_\bullet R \to R with RR on the right regarded as a constant simplicial object is a resolution of RR.

Forming degreewise the module of Kähler differentials on this yields the simplicial object Ω k(P R)\Omega_{k}(P_\bullet R), which is a P RP_\bullet R-module.

Proposition

The cotangent complex of RR is equivalent to

(𝕃Ω /k)(R)R P RΩ k(P R). (\mathbb{L} \Omega_{/k}) (R) \simeq R \otimes_{P_\bullet R} \Omega_k(P_\bullet R) \,.
Remark

Notice that the universal property of the Kähler differentials is that for RR a ring and NN an RR-module, we have

Hom(Ω(R),N)Der(R,N). Hom(\Omega(R), N) \simeq Der(R,N) \,.

Accordingly, it follows that the Andre-Quillen cohomology of RR with values in NN, which is the cohomology of the cosimplicial object

Hom((𝕃Ω)(R),N) Hom((\mathbb{L}\Omega)(R), N)

is equivalently the cohomology of the object

Hom(R P RΩ k(P R),N) Hom P R(Ω k(P R),N) Der(P R,N). \begin{aligned} \cdots & \simeq Hom(R \otimes_{P_\bullet R} \Omega_k(P_\bullet R), N) \\ & \simeq Hom_{P_\bullet R}( \Omega_k(P_\bullet R), N) \\ & \simeq Der(P_\bullet R, N) \end{aligned} \,.

In particular we have the the degree-0 cohomology of this complex is the module of ordinary derivations

H 0(Der(P R,N))Der(R,N). H^0(Der(P_\bullet R, N)) \simeq Der(R,N) \,.
Proposition

If in the above kk is field of characteristic 0, then Andre-Quillen cohomology of the kk-algebra RR with coefficients in a module NN is a direct summand of the corresponding Hochschild cohomology:

H q(Hom(𝕃Ω(R)),N)HH (1) q+1(R,N), H^q(Hom(\mathbb{L} \Omega (R)), N) \simeq HH^{q+1}_{(1)}(R,N) \,,

where the subscript refers to Hodge decomposition of Hochschild cohomology.

This is in section 8.8 of

(,1)(\infty,1)-categorical description

The cotangent complex is a generalization to higher category theory and higher algebra of the notion of cotangent bundle in the sense of Kähler differentials.

Recall from above that for C=C = CRing the ordinary category of commutative rings, the cotangent complex functor is the section

Ω K:RingMod \Omega_K : Ring \to Mod

of the canonical bifibration ModRingMod \to Ring of modules over rings that is on objects given by forming the module of Kähler differentials.

This generalizes to the case where CRing is replaced by any (∞,1)-category CC: the cotangent complex functor for CC is here the left adjoint section

Ω:CT C \Omega : C \to T_C

of the tangent (∞,1)-category projection dom:T CCdom : T_C \to C.

In particular, when C=...C = ..., then the cotangent complex assigns … .

Further properties and applications

For more background see deformation theory.

Apart from simplicial rings we can consider E E_\infty-rings. A map of connective E E_\infty-rings is an equivalence, if it induces an isomorphism at the level of π 0\pi_0 plus a condition on the relative cotangent complex. Similarly, one can express the descent properties of higher stacks via the usual gluing at the bottom level plus the obstruction theory for relative cotangent complex. Study of an appropriate version of the Postnikov tower is a systematic way to do this.

References

See also deformation theory and references therein.

A short exposition (from the point of view of formal schemes) is in

  • chapter 5 (5.29-5.31) in Luc Illusie, Grothendieck’s existence theorem in formal geometry, in FGA explained (179–233) MR2223409; (draft version pdf)

The cotangent complex for a general algebra over an operad in chain complexes is discussed in section 7 of

Revised on December 16, 2014 20:44:37 by Adeel Khan (127.0.0.1)