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Logarithmic geometry and algebraic stacks. (English) Zbl 1069.14022
This paper deals with logarithmic geometry (or logarithmic spaces) in the sense of K. Kato [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191–224 (1989; Zbl 0776.14004)]. Throughout the paper, in which the author assumes that the reader is familiar with the basics of this conceptual framework [L. Illusie, in: Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspect. Math. 15, 183–204 (1994; Zbl 0832.14015)], a log structure on a scheme \(X\) means a log structure on the étale site \(X_{\text{ét}}\) of \(X\) in the sense of Kato.
In this context, the purpose of the present paper is to introduce a stack-theoretic approach to Kato’s theory of logarithmic structures. More precisely, for any fine log scheme \(S\) with underlying scheme \(\mathring S\), the author constructs a fibred category \(\text{Log}_S\to (\mathring S\)-schemes). His main result consists in the proof of the fundamental theorem stating that \(\text{Log}_S\) is an algebraic stack locally of finite presentation over the underlying scheme \(\mathring S\). Moreover, it is shown that a morphism of fine log schemes \(f:X\to S\) defines tautologically a morphism \(\text{Log}(f) : \text{Log}_X\to \text{Log}_S\) of algebraic stacks, and that the association \(S\mapsto \text{Log}_S\) defines a 2-functor from the category of log schemes to the 2-category of algebraic stacks. It is then explained how this 2-functor can be used to reinterpret and study some original basic notions in Kato’s logarithmic geometry.
The fine analysis carried out in this paper is enhanced by an appendix, in which the author compares the notions of log structure in the fppf, étale, and Zariski topology, respectively. Part of this comparison is used in the course of the main body of the paper, and the rest is included for the sake of completeness.
As the author points out, his main theorem on the structure of the stack \(\text{Log}_S\) has further applications which are not discussed in the present paper. Namely, one can develop the theory of log crystalline cohomology using a theory of crystalline cohomology over stacks [cf. M. C. Olsson, Crystalline cohomology of schemes over algebraic stacks, Preprint 2002], and also the deformation theory of log schemes can be analyzed using the structure of \(\text{Log}_S\). In addition, the main theorem of the present paper has a natural place in the moduli theory of fine log schemes [cf. M. C. Olsson, Tohoku Math. J., II. Ser. 55, No. 3, 397–438 (2003; Zbl 1069.14015)]. The author intends to discuss these subjects more thoroughly in forthcoming papers.

MSC:
14D20 Algebraic moduli problems, moduli of vector bundles
14A20 Generalizations (algebraic spaces, stacks)
14A15 Schemes and morphisms
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
18D30 Fibered categories
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