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 Ext1Ext1 group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f0:X0→Y0f0:X0→Y0 of noetherian formal schemes and a closed immersion Y0↪YY0↪Y given by a square zero ideal II, we prove that the set of isomorphism classes of smooth formal schemes lifting X0X0 over YY is classified by Ext1(Ω̂X0/Y01,f0∗I) and that there exists an element in Ext2(Ω̂X0/Y01,f0∗I) which vanishes if and only if there exists a smooth formal scheme lifting X0X0 over YY.