## Abstract

In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata's compactification theorem.

Original language | English |
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Pages (from-to) | 1-42 |

Number of pages | 42 |

Journal | Israel Journal of Mathematics |

Volume | 185 |

Issue number | 1 |

DOIs | |

State | Published - Oct 2011 |

### Bibliographical note

Funding Information:Acknowledgments. I want to express my deep gratitude to B. Conrad for pointing out various gaps and mistakes in an earlier version of the article and to thank R. Huber for a useful discussion. Also I thank D. Rydh and the referee for pointing out some mistakes in §2.3. A first version of the article was written during my stay at the Max Planck Institute for Mathematics at Bonn. The final revision was made when the author was staying at IAS and supported by NFS grant DMS-0635607.