On log flat descent
type: Item
instance of: scholarly article
title: On log flat descent
zbMATH Open document ID: 1275.14005
author: Luc Illusie
author: Chikara Nakayama
author: Takeshi Tsuji
published in: Proceedings of the Japan Academy. Series A
publication date: 2013-05-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1357568331
review text: This paper proves two results that were announced by \textit{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 [\textit{M. C. Olsson}, Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 747--791 (2003; Zbl 1069.14022)], further interpreted in [\textit{K. Kato} and \textit{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).
Mathematics Subject Classification ID: 14B25
zbMATH DE Number: 6164842
zbMATH Keywords: log schemes
zbMATH Keywords: flat descent
zbMATH Keywords: log flat morphism
zbMATH Keywords: log smooth morphism
MaRDI profile type: Publication
OpenAlex ID: W2025182151
cites work: Q3023666
cites work: Logarithmic structures of Fontaine-Illusie. II
cites work: On the conductor formula of Bloch
cites work: Logarithmic étale cohomology
cites work: Quasi-sections in log geometry
cites work: Logarithmic geometry and algebraic stacks
DOI: 10.3792/PJAA.89.1
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description: scientific article; zbMATH DE number 6164842