Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes |
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| Authors: | Leovigildo Alonso Tarrío Ana Jeremías López Marta Pérez Rodríguez |
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| Institution: | 1. Department of Algebra, Faculty of Mathematics , University of Santiago of Compostela , Santiago de Compostela, Spain leoalonso@usc.es;3. Department of Algebra, Faculty of Mathematics , University of Santiago of Compostela , Santiago de Compostela, Spain;4. Department of Mathematics , Superior School of Informatic Engineering, University of Vigo , Ourense, Spain |
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| Abstract: | This a first step to develop a theory of smooth, étale, and unramified morphisms between Noetherian formal schemes. Our main tool is the complete module of differentials, which is, a coherent sheaf whenever the map of formal schemes is of pseudofinite 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 article closes with a version of Zariski's Jacobian criterion. |
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| Keywords: | Formal schemes Infinitesimal lifting Smooth morphisms |
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