Wild ramification and K(pi, 1) spaces

Abstract

We prove that every connected affine scheme of positive characteristic is a K(pi, 1) space for the etale topology. The main ingredient is the special case of the affine space 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.

32 pages, 1 figure; Corrected a mistake in the proof of Theorem 1.3.1 found by an anonymous referee. Numerous minor changes. To appear in Inventiones Mathematicae

References in corpus (2)

• The characteristic cycle and the singular support of a constructible sheaf
• Anabelian geometry with etale homotopy types

Cited by in corpus (5)

• The tame site of a scheme
• The moduli space of cyclic covers in positive characteristic
• $a$-Numbers of Cyclic Degree $p^2$ Covers of the Projective Line
• Spectral decomposition of genuine cusp forms over global function fields
• The derived category of the abelian category of constructible sheaves