Nagata compactification for algebraic spaces

TL;DR

This paper extends the Nagata compactification theorem to separated finite type maps between algebraic spaces, broadening its applicability beyond schemes, and introduces absolute noetherian approximation in this context.

Contribution

It generalizes Nagata compactification and absolute noetherian approximation from schemes to algebraic spaces, filling a significant gap in algebraic geometry.

Findings

Proves Nagata compactification for algebraic spaces.

Establishes absolute noetherian approximation for algebraic spaces.

Extends earlier results from schemes to algebraic spaces.

Abstract

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.