  We make a systematic study of the infinitesimal lifting conditions of a
pseudo finite type map of noetherian formal schemes. We recover the usual
general properties in this context, and, more importantly, we uncover some new
phenomena. We define a completion map of formal schemes as the one that arises
canonically by performing the completion of a noetherian formal scheme along a
subscheme, following the well-known pattern of ordinary schemes. These maps are
etale in the sense of this work (but not adic). They allow us to give a local
description of smooth morphisms. This morphisms can be factored locally as a
completion map followed from a smooth adic morphism. The latter kind of
morphisms can be factored locally as an etale adic morphism followed by a
(formal) affine space. We also characterize etale adic morphisms giving an
equivalence of categories between the category of etale adic formal schemes
over a noetherian formal scheme (X, O_X) and the category of etale schemes over
the ordinary scheme (X, O_X/I), with I an Ideal of definition. These results
characterize completely the local behavior of formally smooth pseudo finite
type (i.e. smooth) maps of noetherian formal schemes. In the final chapter, we
study the basic deformation theory of smooth morphisms.
