# Wild ramification and K(pi, 1) spaces

## TL;DR

This paper proves that connected affine schemes in positive characteristic are K(pi, 1) spaces for the etale topology, using induction, ramification theory, and recent advances in higher ramification theory.

## Contribution

It establishes the K(pi, 1) property for affine schemes in positive characteristic, including rigid analytic and mixed characteristic cases, with new ramification results.

## Key findings

- Connected affine schemes are K(pi, 1) spaces in positive characteristic.
- A key Bertini-type statement on wild ramification of l-adic local systems.
- Extensions to rigid analytic and mixed characteristic settings.

## 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.