$K({\itπ}, 1)$-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 $K({\itπ}, 1)$ -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 $K({\itπ}, 1)$ 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.