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Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes

Leovigildo Alonso·Ana Jeremias·Marta Perez
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Abstract

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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Leovigildo Alonso, Ana Jeremias, Marta Perez. 2006-04-11. Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes. https://arxiv.org/abs/math/0604241

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