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$K({\it\pi},1)$-neighborhoods and comparison theorems
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.
