A characteristic $p$ analog of formal lifting properties

Rankeya Datta; Noah Olander

Paper

A characteristic $p$ analog of formal lifting properties

Abstract

A field extension

of characteristic

is formally étale if and only if the relative Frobenius of

is an isomorphism. Inspired by this classical result, we explore whether the formally étale property for a map

-algebras is characterized by isomorphism of the relative Frobenius

. While

being an isomorphism implies

is formally étale, the converse fails in the non-Noetherian setting. Thus, following Morrow, we introduce an enhancement of the formally étale property that we call b-nil (bounded nil) formally étale, and we show that

is an isomorphism precisely when

is b-nil formally étale. We prove this result by first establishing several structural properties of b-nil formally smooth maps, which are defined analogously to the formally smooth case. Our structural results reveal that the b-nil formally smooth (resp. étale) property is quite different from the formally smooth (resp. étale) property. For instance, we show that any b-nil formally smooth algebra over an

-pure ring is reduced, whereas non-reduced formally étale algebras exist over

by a construction of Bhatt. We also show that the b-nil formally étale property neither implies nor is implied by having a trivial cotangent complex. We explore when formally smooth (resp. étale) implies b-nil formally smooth (resp. étale) in prime characteristic. A satisfactory picture emerges for ideal adic completions.