Logarithmic Flatness

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A map of fine log schemes X → Y induces a map from the scheme underlying X to Olsson's algebraic stack of strict morphisms of fine log schemes over Y. A sheaf on X is called log flat over Y iff it is flat over this algebraic stack. This paper is a study of log flatness and the related notions of flatness for maps of monoids and graded rings. It is shown that log flatness is equivalent to a more general notion of "formal log flatness" that makes sense for an arbitrary map of log ringed topoi. Concrete log flatness criteria are given for many X → Y that occur "in nature," such as toric varieties, nodal curves, and the like. For very simple X → Y it turns out that log flatness is equivalent to previously extant notions of "perfection," thus it provides a generalization for more complicated X → Y useful for studying moduli of sheaves via degeneration techniques.