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Published online by Cambridge University Press: 02 November 2018
Chapter IV is devoted to the key concept of smoothness in log geometry.It begins with a study of log derivations and differentials and relates them to deformation theory.The framework of Grothendieck then leads to a very natural definition of smooth, étale, and unramified morphisms.Although more complicated than in the classical case, the localstructure of such morphisms can be made fairly explicit, and there are differential criteria for smoothness of a morphism. The relationshipof smoothness to regularity is discussed, along with the opennes of the regular locus.Smoothness is also related to flatness, whose definition in the log context is quite subtle. The chapter concludes with a study of the structure of integral, exact, and saturated morphisms in the flat and the smooth cases.
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