Local structure theorems for smooth maps of formal schemes

Details Local structure theorems for smooth maps of formal schemes Local structure theorems for smooth maps of formal schemes

2006-05-04

arxiv.org/abs/math/0605115v3

Mathematics

Algebraic Geometry

This paper, together with math.AG/0604241 supersedes chapters 1-3 of math.AG/0504256. Version 2: minor changes. Version 3: 42

Scientific paper

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in math.AG/0604241. 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 etale 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 etale adic morphism.

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