Compactification for essentially finite-type maps

TL;DR

This paper proves that separated essentially finite-type maps of noetherian schemes can be factored into an injective localization followed by a proper finite-type map, extending classical results and applications in algebraic geometry.

Contribution

It introduces a new factorization theorem for essentially finite-type maps, generalizing Nagata's compactification and providing streamlined versions of key geometric results.

Findings

Any such map factors as an injective localization followed by a proper finite-type map.

Applications include simplified proofs of Zariski's Main Theorem and Chow's Lemma.

Provides essentialized versions of classical algebraic geometry theorems.

Abstract

We show that any separated essentially finite-type map $f$ of noetherian schemes globally factors as $f = hi$ where $i$ is an injective localization map and $h$ a separated finite-type map. In particular, via Nagata's compactification theorem, $h$ can be chosen to be proper. We apply these results to Grothendieck duality. We also obtain other factorization results and provide essentialized versions of many general results such as Zariski's Main Theorem, Chow's Lemma, and blow-up descriptions of birational maps.