nLab
cotangent complex

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.

Quillen’s definition

Using the module of Kähler differentials is not appropriate in general; instead we need to take its derived version. To talk about the nonabelian derived functors, Quillen introduced a model category structure on the category of simplicial commutative rings. Given a morphism $f:A\to B$ of rings, which makes $B$ an $A$-algebra, the category $\mathrm{AbGr}(A$-$\mathrm{Alg}/B)$ of abelian group objects in the slice category $A$-$\mathrm{Alg}/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules. In particular we can consider the forgetful functor $\mathrm{AbGr}(A$-$\mathrm{Alg}/B)\to A$-$\mathrm{Alg}/B$ which has a left adjoint ${\mathrm{Ab}}_{B/A}:A$-$\mathrm{Alg}/B\to \mathrm{AbGr}(A$-$\mathrm{Alg}/B)\cong {}_B\mathrm{Mod}$. All said is true for simplicial commutative rings as well. Now the relative cotangent complex $L_{B/A}$ is the value on $B$ of the left derived functor $𝕃{\mathrm{Ab}}_{B/A}(B)$. Regarding that the left adjoint at the nonderived level (and for usual rings) can be expressed via Kähler differentials, this explains the phrase “derived version of the module Kähler differentials”.

$(\mathrm{\infty }\mathrm{,1})$-categorical viewpoint

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.

For $C=$ Ring the ordinary category of commutative rings, the cotangent complex functor is the section

$${\Omega }_K:\mathrm{Ring}\to \mathrm{Mod}$$\Omega_K : Ring \to Mod

of the canonical bifibration $\mathrm{Mod}\to \mathrm{Ring}$ of modules over rings that is on objects given by forming the module of Kähler differentials.

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

$$\Omega :C\to T_C$$\Omega : C \to T_C

of the tangent (∞,1)-category projection $T_C\to C$.

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

Further properties and applications

For more background see deformation theory.

Apart from simplicial rings we can consider $E_{\mathrm{\infty }}$-rings. A map of connective $E_{\mathrm{\infty }}$-rings is an equivalence, if it induces an isomorphism at the level of ${\pi }_0$ 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

• Alexander Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lec. Notes in Math. 79

• L. Illusie, Complexe cotangent et déformations I, Lec. Notes Math. 239, Springer 1971, xv+355 pp.; II, LNM 283, Springer 1972. vii+304 xv+355 pp.

• Kai Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88, MR1437495 (98e:14022) arXiv:alg-geom/9601010

• B. Fantechi, M. Manetti, Obstruction calculus for functors of Artin rings I, J. Algebra 202 (1998), no. 2, 541–576, MR1617687 (99f:14004).

• Jacob Lurie, Deformation Theory, (arXiv:0709.3091)

• S. Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Alg. 120, 77–104 (1997) (doi)

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)