Cotangent complex.html

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 $f\colon A\rightarrow B$ is a morphism in M. The cotangent complex $L f$ (or $L B / A$ ) is an object in the category of spectra in $M B / / B$ . A pair of composable morphisms $A\xrightarrow{f} B\xrightarrow{g} C$ induces an exact triangle in the homotopy category, $L_{B/A}\otimes_BC\rightarrow L_{C/A}\rightarrow L_{C/B}\rightarrow (L_{B/A}\otimes_BC)[-1]$ .

In the special case where $B$ is an algebra over a ring $A$ , the cotangent complex is constructed as follows: One finds a resolution $P^{\bullet} \to B$ of $B$ by simplicial free $A$ algebras. Then one applies the functor of Kahler differentials to $P^{\bullet}$ . The cotangent complex $L B / A$ is the complex associated to the resulting simplicial object.

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