ON SMOOTH, UNRAMIFIED, ETALE AND FLAT MORPHISMS OF FINE LOGARITHMIC SCHEMES

The notion of unramified morphisms of schemes is generalized in a natu ral way to the category of fine logarithmic schemes. There are given s everal equivalent conditions for a morphism of fine log schemes to be unramified: vanishing of the differential module, all fibres to be unr amified and a local structure theorem using charts of the log structur es. 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 sens e of K. KATO. Let f: X --> Y be a map of fine log schemes. Then f is s mooth if and only if locally it can be factorized over an etale map in to the standard log affine space over Y: The map f is etale if and onl y if it is flat and unramified. Further there are generalizations of t he usual fibre criterions for flatness or smoothness of morphisms of s chemes to the context of log schemes: Let f: X --> Y be an S-morphism of fine log schemes. Assume that X/S is hat and that the underlying ma ps of schemes are locally of finite presentation. Then f is flat if an d only if the induced maps on the fibres f(s):X(s) --> Y-s, s is an el ement of S, are flat. Finally f is smooth if and only if it is hat and the induced maps on the fibres f(-1)(y)) --> y, y is an element of Y, are smooth.