Basic deformation theory of smooth formal schemes

We provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341–1367]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext1 group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism image of noetherian formal schemes and a closed immersion image given by a square zero ideal image, we prove that the set of isomorphism classes of smooth formal schemes lifting image over image is classified by image and that there exists an element in image which vanishes if and only if there exists a smooth formal scheme lifting image over image.