Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes

Details Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes

2006-04-11

arxiv.org/abs/math/0604241v1

Mathematics

Algebraic Geometry

26 pages

Scientific paper

This a first step to develop a theory of smooth, etale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of noetherian formal schemes is flat and its module of differentials is locally free. The paper closes with a version of Zariski's Jacobian criterion.

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