Tohoku Mathematical Journal
2004
March
SECOND SERIES VOL. 56, NO. 1

Tohoku Math. J.
56 (2004), 79-104

Title COMPACTIFICATIONS OF LOG MORPHISMS

Author Elmar Grosse-Klönne

(Received March 4, 2002, revised April 22, 2003)
Abstract. We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.

2000 Mathematics Subject Classification. Primary 14F40; Secondary 14F30.
Key words and phrases. Logarithmic structures, de Rham cohomology, crystalline cohomology, semistable reduction.

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