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On log flat descent. (English) Zbl 1275.14005

This paper proves two results that were announced by K. Kato [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191–224 (1989; Zbl 0776.14004)]. The first result is that for \(f: X \rightarrow Y\) a morphism of fine, separated (fs for short) log schemes, \(g:Y' \rightarrow Y\) a surjective, Kummer, log flat morphism of fs log schemes locally of finite presentation, \(f': X \times_Y Y' \rightarrow Y'\) the morphism induced by \(f\), then \(f\) is log étale (resp. log smooth, log flat) if and only if \(f'\) is log étale (resp. log smooth, log flat). The second result is that if \(X \overset{f} \rightarrow Y \overset {g} \rightarrow Z\) are morphisms of fs log schemes, where \(f\) is surjective and Kummer, then if both \(f\) and \(g \circ f\) are log étale (resp. log smooth, log flat) then \(g\) is log étale (resp. log smooth, log flat). The authors prove both results by making use of a characterization of the properties of morphism of fs log schemes of being log étale, log smooth, log flat in the language of stacks [M. C. Olsson, Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 747–791 (2003; Zbl 1069.14022)], further interpreted in [K. Kato and T. Saito, Publ. Math., Inst. Hautes Étud. Sci. 100, 5–151 (2004; Zbl 1099.14009)] as saying that a morphism \(f:X \rightarrow Y\) of fs log schemes is log étale (resp. log smooth, log flat) if and only if for any \(Y' \rightarrow Y\) morphism of fs log schemes, for any \(X'' \rightarrow X \times_Y Y' = X'\) log étale morphism of fs log schemes for which \(X'' \rightarrow X' \rightarrow Y'\) is strict, then the underlying morphism of this composite is étale (resp. smooth, flat).

MSC:

14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
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References:

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[8] M. C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747-791. · Zbl 1069.14022 · doi:10.1016/j.ansens.2002.11.001
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