Cotangent Complex - The Definition of The Cotangent Complex

The Definition of The Cotangent Complex

The correct definition of the cotangent complex begins in the homotopical setting. Quillen and André worked with the simplicial commutative rings, while Illusie worked with simplicial ringed topoi. For simplicity, we will consider only the case of simplicial commutative rings. Suppose that A and B are simplicial rings and that B is an A-algebra. Choose a resolution r : P• → B of B by simplicial free A-algebras. Applying the Kähler differential functor to P• produces a simplicial B-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 Ω_B/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 LB/A ⊗_B D → LD/C which respects the augmentation maps. This map is constructed by choosing a free simplicial C-algebra resolution of D, say s : Q• → D. Because P• is a free object, the composite hr can be lifted to a morphism P• → Q•. Applying functoriality of Kähler differentials to this morphism gives the required morphism of cotangent complexes. In particular, given homomorphisms A → B → C, 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 induces an exact triangle in the homotopy category, .