Basic deformation theory of smooth formal schemes

Abstract

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 Ext 1 group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f 0 : X 0 → Y 0 of noetherian formal schemes and a closed immersion Y 0 ↪ Y given by a square zero ideal I , we prove that the set of isomorphism classes of smooth formal schemes lifting X 0 over Y is classified by Ext 1 ( Ω ̂ X 0 / Y 0 1 , f 0 ∗ I ) and that there exists an element in Ext 2 ( Ω ̂ X 0 / Y 0 1 , f 0 ∗ I ) which vanishes if and only if there exists a smooth formal scheme lifting X 0 over Y .