$K({\it\pi}, 1)$-neighborhoods and comparison theorems. (5th June 2015)

• Record Type: Journal Article
• Authors: Achinger, Piotr
• Is Part Of: Compositio mathematica. Volume 151:Number 10(2015)
• Page Start: 1945
• Page End: 1964
• Publication Date: 2015-06-05
• Subjects: 14F20 (primary), -- 14F30 (secondary); Faltings' topos, -- étale cohomology, -- logarithmic geometry, -- semistable reduction, -- p-adic Hodge theory, -- K (𝜋, 1) schemes; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X15007319
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : 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.