A note on quasi-perfect morphisms

TL;DR

This paper investigates quasi-perfect morphisms between Noetherian algebraic spaces, demonstrating their local detectability at étale local rings and establishing the openness of the quasi-perfect locus.

Contribution

It provides new criteria for detecting quasi-perfectness locally and proves the Zariski openness of the quasi-perfect locus for proper morphisms.

Findings

Quasi-perfectness can be detected at étale local rings and their completions.

The locus of quasi-perfect points is Zariski open.

Local behavior of quasi-perfect morphisms is characterized in detail.

Abstract

This note is concerned with quasi-perfect morphisms between Noetherian algebraic spaces. In particular, we study the local behavior of quasi-perfect proper morphisms. We show that quasi-perfectness of a proper morphism can be detected at the \'{e}tale local rings of points of the target, as well as their completions and (strict) Henselizations. As a corollary, we obtain that the locus of points where a proper morphism is quasi-perfect is Zariski open.