## Abstract

We strengthen Gabber's l'-alteration theorem by avoiding all primes in- vertible on a scheme. In particular, we prove that any scheme X of finite type over a quasi-excellent threefold can be desingularized by a char(X)- alteration, i.e., an alteration whose order is only divisible by primes nonin- vertible on X. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of X can be split into a composition of a tame Galois alteration and a char(X)-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field k of residue characteristic p has no nontrivial p-extensions, then any algebraic extension l/k is tame.

Original language | American English |
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Pages (from-to) | 97-126 |

Number of pages | 30 |

Journal | Annals of Mathematics |

Volume | 186 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Department of Mathematics, Princeton University.

## Keywords

- Alterations
- Resolution of singularities
- Tame distillation
- Valuations