K(π, 1)-neighborhoods and comparison theorems. (5th June 2015) - Achinger, Piotr
Abstract : A technical ingredient in Faltings' original approach to p -adic comparison theorems involves the construction of K(π, 1) -neighborhoods for a smooth scheme X over a mixed characteristic discrete valuation ring with a perfect residue field: every point xX has an open neighborhood U whose generic fiber is a K(π, 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.