Prüfer algebraic spaces

Details Prüfer algebraic spaces Prüfer algebraic spaces

2011-01-17

arxiv.org/abs/1101.3199v1

Mathematics

Algebraic Geometry

34 pages. First version, comments are welcome

Scientific paper

This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Pr\"ufer pairs of algebraic spaces that generalize spectra of Pr\"ufer rings, and pushouts of algebraic spaces that generalize composition of valuations and Ferrand's pincements of schemes. As a particular case of Pr\"ufer spaces we introduce valuation algebraic spaces, and use them to establish valuative criteria of properness and separatedness that sharpen the standard criteria. In the sequel paper, we will introduce a version of Riemann-Zariski spaces (RZ spaces), and will prove Nagata compactification theorem for algebraic spaces.

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