A characteristic $p$ analog of formal lifting properties

A field extension $L/K$ of characteristic $p > 0$ is formally étale if and only if the relative Frobenius of $L/K$ is an isomorphism. Inspired by this classical result, we explore whether the formally étale property for a map $R \to S$ of $\mathbf{F}p$-algebras is characterized by isomorphism of the relative Frobenius $F{S/R}$. While $F_{S/R}$ being an isomorphism implies $R \to S$ 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 $F_{S/R}$ is an isomorphism precisely when $R \to S$ 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 $F$-pure ring is reduced, whereas non-reduced formally étale algebras exist over $\mathbf{F}_p$ 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.

💡 Research Summary

This paper, “A characteristic p analog of formal lifting properties,” conducts a deep investigation into the relationship between the formal lifting properties of ring homomorphisms in characteristic p > 0 and the behavior of the relative Frobenius morphism.

The study is motivated by a classical result: for an extension of fields L/K of characteristic p, L/K is formally étale if and only if its relative Frobenius F_{L/K} is an isomorphism. The authors explore whether this characterization extends to maps φ: R → S of general F_p-algebras. They confirm that if F_φ is an isomorphism, then φ is formally étale. However, they provide counterexamples demonstrating that the converse fails in the non-Noetherian setting. This discrepancy arises because the standard definition of formally étale only requires lifting conditions for nilpotent ideals I with I^2 = 0, which is insufficient to control the relative Frobenius.

To bridge this gap, the authors introduce a refined notion called “b-nil formally étale” (where “b-nil” stands for bounded nilpotent). For a map φ: R → S, this property requires that for every test pair (A, I) where I is an ideal with I^