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An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. (English) Zbl 1052.14005

Berthelot, Pierre (ed.) et al., \(p\)-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France (ISBN 2-85629-117-1/pbk). Astérisque 279, 271-322 (2002).
Summary: This paper is a report on the work of K. Fujiwara, K. Kato and C. Nakayama on log étale cohomology of log schemes. After recalling basic terminology and facts on log schemes we define and study a class of log étale morphisms of log schemes, called Kummer étale morphisms, which generalize the tamely ramified morphisms of classical algebraic geometry. We discuss the associated topology and cohomology. The main results are comparison theorems with classical étale cohomology and log Betti cohomology, a theorem of invariance of Kummer étale cohomology under log blow-ups (for which we provide a complete proof) and a local acyclicity theorem for log smooth log schemes over the spectrum of a henselian discrete valuation ring, which implies tameness for the corresponding classical nearby cycles. In the last section we state results of K. Kato on log étale cohomology, where localization by Kummer étale morphisms is replaced by localization by all log étale morphisms.
For the entire collection see [Zbl 0990.00020].

MSC:

14A99 Foundations of algebraic geometry
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D10 Arithmetic ground fields (finite, local, global) and families or fibrations
14E22 Ramification problems in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
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