Algebraic Geometry
A Note on Kato–Nakayama spaces
Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 261-264.

In this Note we prove the following result. A fine log scheme over the complex numbers and its saturated have homeomorphic Kato–Nakayama associated spaces. Moreover these spaces are isomorphic as ringed spaces, either with the ring sheaf defined by Kato–Nakayama, or with that defined by Ogus. In the definition of these spaces, non-integral monoids are involved, so that the proof of the result is based on properties of nonnecessarily integral monoids.

Dans cette Note nous prouvons le resultat suivant. Un log schéma sur le corps des nombres complexes et son saturé ont des espaces de Kato–Nakayama associés qui sont homéomorphes. En plus, ces espaces sont isomorphes en tant qu'espaces annelés, soit avec le faisceau d'anneaux défini par Kato–Nakayama, soit avec le faisceau d'anneaux défini par Ogus. Dans la définition de ces espaces on utilise des monoïdes non intègres, et la démonstation utilise certaines proprietés des monoïdes non nécessairement intègres.

Published online:
DOI: 10.1016/j.crma.2004.06.009
Cailotto, Maurizio 1

1 Dip. di Matematica, Via Belzoni 7, 35131 Padova, Italy
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     title = {A {Note} on {Kato{\textendash}Nakayama} spaces},
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Cailotto, Maurizio. A Note on Kato–Nakayama spaces. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 261-264. doi : 10.1016/j.crma.2004.06.009.

[1] Kato, K. Logarithmic structures of Fontaine–Illusie, Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, 1989, pp. 191-224

[2] Kato, K.; Nakayama, C. Log Betti cohomology, log étale cohomology and log de Rham cohomology of log schemes over C, Kodai Math. J., Volume 22 (1999), pp. 161-186

[3] Illusie, L. An overview of the work of K. Fujiwara, K. Kato and C. Nakayama on logarithmic etale cohomology, Astérisque, Volume 279 (2002), pp. 271-322

[4] Lorenzon, P. Indexed algebras associated to a log structure and a theorem of p-descent on log schemes, Manuscripta Math., Volume 101 (2000), pp. 271-299

[5] A. Ogus, Logarithmic De Rham cohomology, Preprint 1997

[6] Ogus, A. On the logarithmic Riemann–Hilbert correspondence, Documenta Math. (2003), pp. 655-724 (Extra Volume: Kazuya Kato's Fiftieth Birthday)

[7] T. Tsuji, Saturated morphisms of logarithmic schemes, Preprint, 1997

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