Local structure theorems for smooth maps of formal schemes

Abstract

Communicated by E.M. Friedlander MSC:Primary: 14B10 secondary: 14B20 14B25 a b s t r a c tWe continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007Alg. 35 ( ) 1341Alg. 35 ( -1367. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism. IntroductionFormal schemes have always been present in the backstage of algebraic geometry but they were rarely studied in a systematic way after the foundational [5, Section 10]. It has become more and more clear that the wide applicability of formal schemes in several areas of mathematics require such study. Let us cite a few of this applications. The construction of De Rham cohomology for a scheme X of zero characteristic embeddable in a smooth scheme P, studied by Hartshorne [9] (and, independently, by Deligne), is defined as the hypercohomology of the completion of the De Rham complex of the formal completion of P along X . Formal schemes play a key role in p-adic cohomologies (crystalline, rigid . . . ) and are also algebraic models of rigid analytic spaces. These developments go back to Grothendieck with further elaborations by Raynaud, in collaboration with Bosch and Lütkebohmert, and later work by Berthelot and de Jong. In a different vein, Strickland [15] has pointed out the importance of formal schemes in the context of (stable) homotopy theory.A particular assumption that it is almost always present in most earlier works on formal schemes is that morphisms are adic, i.e. that the topology of the sheaf of rings of the initial scheme is induced by the topology of the base formal scheme. This hypothesis on a morphism of formal schemes guarantees that its fibers are usual schemes, therefore an adic morphism between formal schemes is, in the terminology of Grothendieck's school, a relative scheme over a base that is a formal scheme. But there are important examples of maps of formal schemes that do not correspond to this situation. The first example that comes into mind is the natural map Spf(A[[X ]]) → Spf(A) for an adic ring A. This morphism has a finiteness $ This work was partially supported by Spain's MCyT and E.U.'s FEDER research project MTM2005-05754.

Hypertoric Hitchin systems and Kirchhoff polynomials

“…In the following we denote by Spf A the affine formal scheme associated to a complete local ring. We refer the reader to [TLR09] for definition and properties of smooth morphisms of formal schemes.…”