On Smooth, Unramified, Étale and Flat Morphisms of Fine Logarithmic Schemes
✍ Scribed by Werner Bauer
- Publisher
- John Wiley and Sons
- Year
- 1995
- Tongue
- English
- Weight
- 579 KB
- Volume
- 176
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
The notion of unramified morphisms of schemes is generalized in a natural way to the category of fine logarithmic schemes. There are given several equivalent conditions for a morphism of fine log schemes to be unramified: vanishing of the differential module, all fibres to be unramified and a local structure theorem using charts of the log structures. In the main part of the paper there are shown some criterions for a morphism of fine log schemes to be smooth, flat or etale in the sense of K. KATO. Let /: X + Y be a map of fine log schemes. Then/ is smooth if and only if locally it can be factorized over an etale map into the standard log afline space over I: The mapfis etale if and only if it is flat and unramified. Further there are generalizations of the usual fibre criterions for flatness or smoothness of morphisms of schemes to the context of log schemes: Let/: X -+ Y be an S-morphism of fine log schemes. Assume that X/S is flat and that the underlying maps of schemes are locally of finite presentation. Then / is flat if and only if the induced maps on the fibres f,: X, + Y,, s E S, are flat. Finally / is smooth if and only if it is flat and the induced maps on the fibres /-l(y) -+ y, y E Y, are smooth.