## Abstract

In this exposé we state and prove Gabber’s modification theorem mentioned in the introduction (see step (C)). Its main application is to Gabber’s refined—i.e., prime to `—local uniformization theorem. This is treated in exposé IX. A relative variant of the modification theorem, also due to Gabber, has applications to prime to `refinements of theorems of de Jong on alterations of schemes of finite type over a field or a trait. This is discussed in exposé X. In § 1, we state Gabber’s modification theorem in its absolute form (Theorem 1.1). The proof of this theorem occupies § 4–5.

A key ingredient is the existence of functorial (with respect to regular morphisms) resolutions in characteristic zero; the relevant material is collected in § 2. We apply it in § 3 to get resolutions of log regular log schemes, using the language of Kato’s fans and Ogus’s monoschemes. The main results, on which the proof of 1.1 is based, are Theorems 3.3.16 and 3.4.15. §2 and 3 can be read independently of § 1, 4, 5.

Though we basically follow the lines of Gabber’s original proof, our approach differs from it at several places, especially in our use of associated points and saturated desingularization towers, whose idea is due to the second author. In 2.3.13 and 2.4

we discuss material from Gabber’s original proof.

A key ingredient is the existence of functorial (with respect to regular morphisms) resolutions in characteristic zero; the relevant material is collected in § 2. We apply it in § 3 to get resolutions of log regular log schemes, using the language of Kato’s fans and Ogus’s monoschemes. The main results, on which the proof of 1.1 is based, are Theorems 3.3.16 and 3.4.15. §2 and 3 can be read independently of § 1, 4, 5.

Though we basically follow the lines of Gabber’s original proof, our approach differs from it at several places, especially in our use of associated points and saturated desingularization towers, whose idea is due to the second author. In 2.3.13 and 2.4

we discuss material from Gabber’s original proof.

Original language | English |
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Title of host publication | Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l'École Polytechnique 2006--2008 |

Publisher | Societe Mathematique de France |

Chapter | 8 |

Pages | 103-160 |

Number of pages | 58 |

ISBN (Print) | 978-2-85629-790-2 |

State | Published - 2014 |

### Publication series

Name | Astérisque |
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Publisher | Société Mathématique de France |

Number | 363-364 |