Deformation of formal schemes through local homology.

We define the cotangent complex of a morphism f : X → Y of locally noetherian formal schemes as an object in the derived category D − ( X ) through local homology. We discuss its basic properties and establish the basics results of a deformation theory, providing a characterization of smooth and étale morphisms. This leads to simpler lifting results depending on a differential module, for a class of non-smooth morphism of usual schemes. We also give descriptions of the cotangent complex in the case of regular closed immersions and complete intersection morphisms of formal schemes. [ABSTRACT FROM AUTHOR]