Nagata embedding and A-schemes

TL;DR

This paper introduces normal and approximable A-schemes, showing their properties and applications, including a new proof of Nagata embedding via Zariski-Riemann spaces, linking scheme theory and compactification.

Contribution

It defines and studies approximable A-schemes, demonstrating their properties and their role in proving Nagata embedding through Zariski-Riemann spaces.

Findings

Approximable A-schemes inherit key properties of schemes

Zariski-Riemann space can be viewed as a limit or compactification

Nagata embedding is proved using Zariski-Riemann spaces

Abstract

We define the notion of normal A-schemes, and approximable A-schemes. Approximable A-schemes inherit many good properties of ordinary schemes. As a consequence, we see that the Zariski-Riemann space can be regarded in two ways -- either as the limit space of admissible blow ups, or as the universal compactification of the given non-proper scheme. We can prove Nagata embedding using Zariski-Riemann spaces.