[Paper Review] Valuations and Frobenius

Rankeya Datta, Karen E. Smith|arXiv (Cornell University)|Jul 21, 2015

Commutative Algebra and Its Applications43 references3 citations

TL;DR

This paper investigates Frobenius singularities—F-purity, Frobenius splitting, and F-regularity—in valuation rings of prime characteristic, showing that all such rings are F-pure and introducing F-pure regularity as a natural generalization of strong F-regularity. The key result is that a valuation ring is F-pure regular if and only if it is Noetherian, and for Noetherian valuation rings in F-finite function fields, Frobenius splitting is equivalent to excellence and F-finiteness.

ABSTRACT

The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.

Motivation & Objective

• To extend the theory of Frobenius singularities—F-purity, splitting, and F-regularity—beyond the Noetherian setting to valuation rings.
• To understand the behavior of the Frobenius map in non-Noetherian valuation rings, especially in function fields.
• To clarify the relationship between Frobenius finiteness, splitting, excellence, and F-finiteness in Noetherian valuation rings.
• To introduce and analyze F-pure regularity as a generalization of strong F-regularity for non-F-finite rings.
• To determine whether Frobenius splitting implies F-finiteness or excellence in non-Noetherian valuation rings.

Proposed method

• Use the Frobenius map $F: R o R$, $r o r^p$, as the central tool to analyze singularities in valuation rings of prime characteristic.
• Define F-pure regularity as a generalization of strong F-regularity using pure maps instead of split maps, avoiding finiteness assumptions.
• Prove that all valuation rings are F-pure using the purity of the Frobenius map, leveraging flatness and valuation-theoretic properties.
• Establish that Frobenius is finite for a valuation ring in a function field if and only if the valuation is Abhyankar, using the structure of value groups.
• Apply the theory of excellence and F-finiteness to characterize Frobenius splitting in Noetherian valuation rings.
• Use examples and counterexamples (e.g., non-Frobenius split DVRs) to distinguish between F-purity, splitting, and stronger regularity conditions.

Experimental results

Research questions

• RQ1 Is the Frobenius map always pure (F-pure) in valuation rings of prime characteristic?
• RQ2 What is the precise relationship between Frobenius splitting, F-finiteness, and excellence in Noetherian valuation rings?
• RQ3 Does F-pure regularity characterize Noetherian valuation rings among all valuation rings?
• RQ4 Can Frobenius splitting occur in non-Noetherian valuation rings, and if so, under what conditions?
• RQ5 Is F-pure regularity equivalent to all modules being tightly closed in the non-Noetherian setting?

Key findings

• All valuation rings of prime characteristic are F-pure, as the Frobenius map is always pure.
• A valuation ring is F-pure regular if and only if it is Noetherian, establishing a sharp dichotomy between Noetherian and non-Noetherian behavior.
• For Noetherian valuation rings in function fields over F-finite fields, Frobenius splitting is equivalent to F-finiteness and to excellence.
• The Frobenius map is finite if and only if the valuation is Abhyankar, and in this case, the valuation ring is Frobenius split.
• There exist Noetherian valuation rings (e.g., discrete valuation rings) that are not Frobenius split, showing that splitting is strictly stronger than F-purity.
• Split F-regularity is strictly stronger than F-pure regularity; for DVRs with F-finite fraction field, it is equivalent to excellence and F-finiteness.