Cotangent complex
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In mathematics the cotangent complex is a roughly a universal linearization of a morphism. Cotangent complexes for morphisms of simplicial commutative rings were first made explicit by Luc Illusie in his PhD thesis.
Suppose that M is a combinatorial model category and 
is a morphism in M. The cotangent complex Lf (or LB / A) is an object in the category of spectra in MB / / B. A pair of composable morphisms 
induces an exact triangle in the homotopy category, ![L_{B/A}\otimes_BC\rightarrow L_{C/A}\rightarrow L_{C/B}\rightarrow (L_{B/A}\otimes_BC)[-1]](http://upload.wikimedia.org/math/4/5/9/45992d1f217b141798e8220dcbbf9b3d.png)
.
In the special case where B is an algebra over a ring A, the cotangent complex is constructed as follows: One finds a resolution 
of B by simplicial free A algebras. Then one applies the functor of Kahler differentials to 
. The cotangent complex LB / A is the complex associated to the resulting simplicial object.
