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Logarithmic Flatness

A map of fine log schemes $X \to Y$ 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 \emph{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 $X \to Y$ that occur "in nature," such as toric varieties, nodal curves, and the like. For very simple $X \to Y$ it turns out that log flatness is equivalent to previously extant notions of "perfection," thus it provides a generalization for more complicated $X \to Y$ useful for studying moduli of sheaves via degeneration techniques.