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$K({\it\pi},1)$-neighborhoods and comparison theorems
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 10 / October 2015
- Published online by Cambridge University Press:
- 05 June 2015, pp. 1945-1964
- Print publication:
- October 2015
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- Article
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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.
