Wild ramification and K(π,1) spaces

We prove that every connected affine scheme of positive characteristic is a K ( π , 1 ) space for the étale topology. The main ingredient is the special case of the affine space A k n over a field k . This is dealt with by induction on n , using a key “Bertini-type” statement regarding the wild rami...

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Bibliographic Details
Published in:Inventiones mathematicae 2017-11, Vol.210 (2), p.453-499
Main Author: Achinger, Piotr
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Online Access:Get full text
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Summary:We prove that every connected affine scheme of positive characteristic is a K ( π , 1 ) space for the étale topology. The main ingredient is the special case of the affine space A k n over a field k . This is dealt with by induction on n , using a key “Bertini-type” statement regarding the wild ramification of ℓ -adic local systems on affine spaces, which might be of independent interest. Its proof uses in an essential way recent advances in higher ramification theory due to T. Saito. We also give rigid analytic and mixed characteristic versions of the main result.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-017-0733-5