$K(\pi, 1)$-neighborhoods and comparison theorems

TL;DR

This paper extends the construction of $K(, 1)$-neighborhoods from smooth schemes to logarithmically smooth schemes, potentially simplifying proofs in p-adic Hodge theory by generalizing a key technical tool.

Contribution

It generalizes the existence of $K(, 1)$-neighborhoods to the logarithmic setting, broadening the applicability of this technique in p-adic comparison theorems.

Findings

Extended $K(, 1)$-neighborhoods to logarithmically smooth schemes.

Provided a new proof technique using a Nagata trick.

Facilitated potential simplifications in p-adic Hodge theory proofs.

Abstract

A technical ingredient in Faltings' original approach to p-adic comparison theorems involves the construction of $K(\pi, 1)$-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 $K(\pi, 1)$ 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.