-neighborhoods and comparison theorems

Abstract

A technical ingredient in Faltings’ original approach to$p$-adic comparison theorems involves the construction of$K({\it\pi},1)$-neighborhoods for a smooth scheme$X$over a mixed characteristic discrete valuation ring with a perfect residue field: every point$x\in X$has an open neighborhood$U$whose generic fiber is a$K({\it\pi},1)$scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.