On weakly \'etale morphisms

TL;DR

This paper characterizes weakly étale morphisms via a lifting property, introduces Henselian descent, and explores properties of weakly étale algebras over regular rings, highlighting their ind-étale nature and lifting limitations.

Contribution

It provides a new characterization of weakly étale morphisms through a lifting property and introduces Henselian descent, advancing understanding of their geometric and algebraic properties.

Findings

Weakly étale morphisms are characterized by a lifting property similar to formally étale morphisms.

Henselian descent theorem is established, analogous to sheaf conditions for the fpqc topology.

Weakly étale algebras over regular rings are ind-étale and may not lift along surjective homomorphisms.

Abstract

We show that the weakly \'etale morphisms, used to define the pro-\'etale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally \'etale morphisms. In order to prove this, we prove a theorem called Henselian descent which is a "Henselized version" of the fact that a scheme defines a sheaf for the fpqc topology. Finally, we show that weakly \'etale algebras over regular rings arising in geometry are ind-\'etale and that weakly \'etale algebras do not always lift along surjective ring homomorphisms.