Relative Riemann-Zariski spaces

TL;DR

This paper introduces relative Riemann-Zariski spaces associated with scheme morphisms, providing new tools for understanding scheme modifications and factorizations, including a novel proof of Nagata's compactification theorem.

Contribution

It generalizes classical Riemann-Zariski spaces to a relative setting and applies these to prove a strong stable modification theorem and a decomposition theorem for scheme morphisms.

Findings

Established that relative RZ spaces can be described as projective limits or valuation spaces.

Proved a strong stable modification theorem for relative curves.

Provided a new proof of Nagata's compactification theorem.

Abstract

In this paper we study relative Riemann-Zariski spaces attached 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 quasi-separated 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.)