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arXiv · 1407.0337

$K(π, 1)$-neighborhoods and comparison theorems

Piotr Achinger
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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 dvr with a perfect residue field: every point of X has an open neighborhood whose general 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.

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Piotr Achinger. 2014-07-01. $K(π, 1)$-neighborhoods and comparison theorems. https://doi.org/10.1112/s0010437x15007319

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