[Paper Review] Global Frobenius Betti numbers and F-splitting ratio

Alessandro De Stefani, Thomas Polstra+1 more|arXiv (Cornell University)|Nov 27, 2018

Commutative Algebra and Its Applications18 references4 citations

TL;DR

This paper extends Frobenius Betti numbers and the F-splitting ratio to non-local, F-finite domains of prime characteristic, proving their existence and establishing global invariants that detect regularity and F-purity. It resolves [DSPY16, Question 4.24] by showing that the global F-signature of a pair $(R,\mathscr{D})$ is positive if and only if the pair is strongly F-regular, without requiring additional assumptions on the Cartier algebra $\mathscr{D}$.

ABSTRACT

We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair $(R,\mathscr{D})$, where $\mathscr{D}$ is a Cartier algebra, is equivalent to the positivity of the global F-signature ${ m s}(R,\mathscr{D})$ of the pair. This extends a result previously proved by these authors, by removing an extra assumption on the Cartier algebra.

Motivation & Objective

• To generalize Frobenius Betti numbers and F-splitting ratio from local to global settings in prime characteristic rings.
• To remove extra assumptions on Cartier algebras required in prior work to characterize strong F-regularity via F-signature.
• To establish global invariants that detect regularity and F-purity in non-local domains.
• To define and analyze the splitting rate and F-splitting ratio for non-local rings using localization and asymptotic limits.

Proposed method

• Define global Frobenius Betti numbers $\beta_i^F(R)$ and Frobenius Euler characteristics $\chi_i^F(R)$ via limits of normalized Tor-lengths over $F^e_*R$.
• Introduce the splitting rate $\operatorname{sr}(R)$ as a global extension of splitting dimension, defined via minimal generators of syzygies.
• Prove existence of the limits $\beta_i^F(R)$, $\chi_i^F(R)$, and $r_F(R)$ using asymptotic analysis and finiteness of $F$-finite rings.
• Use localization techniques to relate global invariants to local invariants, showing $\chi_i^F(R) = \max\{\chi_i^F(R_P)\}$ and $\operatorname{sr}(R) = \min\{\operatorname{sr}(R_P)\}$.
• Establish positivity of $r_F(R)$ as equivalent to F-purity, and $\operatorname{s}(R,\mathscr{D}) > 0$ as equivalent to strong F-regularity of $(R,\mathscr{D})$.
• Apply a key lemma on the existence of a uniform lower bound $\varepsilon > 0$ on the free rank of $F^e_*R$, enabling the positivity of the F-signature limit.

Experimental results

Research questions

• RQ1 Can Frobenius Betti numbers and the F-splitting ratio be meaningfully extended beyond local rings to general F-finite domains in prime characteristic?
• RQ2 Is the global F-signature $\operatorname{s}(R,\mathscr{D})$ positive if and only if the pair $(R,\mathscr{D})$ is strongly F-regular, without additional assumptions on $\mathscr{D}$?
• RQ3 How do global invariants like $\beta_i^F(R)$ and $r_F(R)$ relate to their local counterparts under localization?
• RQ4 Does the splitting rate $\operatorname{sr}(R)$ satisfy a min-property over localizations, and does $r_F(R)$ inherit its positivity from local behavior?
• RQ5 Can the global F-signature be bounded below uniformly when the ring is strongly F-regular, enabling convergence of the limit defining $\operatorname{s}(R,\mathscr{D})$?

Key findings

• The global Frobenius Betti numbers $\beta_i^F(R)$ and Frobenius Euler characteristics $\chi_i^F(R)$ exist for any F-finite domain $R$ of prime characteristic.
• The global Frobenius Euler characteristic satisfies $\chi_i^F(R) = \max\{\chi_i^F(R_P) \mid P \in \operatorname{Spec}(R)\}$, linking global invariants to local ones.
• The ring $R$ is regular if and only if $\beta_i^F(R) = 0$ for some $i > 0$, generalizing a local result to the global setting.
• The F-splitting ratio $r_F(R)$ is positive if and only if $R$ is F-pure, and $r_F(R) = \min\{r_F(R_P) \mid \operatorname{sr}(R) = \operatorname{sr}(R_P)\}$, establishing a min-property for the ratio.
• The global F-signature $\operatorname{s}(R,\mathscr{D})$ is positive if and only if the pair $(R,\mathscr{D})$ is strongly F-regular, resolving [DSPY16, Question 4.24] without extra assumptions on $\mathscr{D}$.
• A uniform lower bound $\varepsilon > 0$ exists such that $a_e(R) \geq \varepsilon \cdot \operatorname{rank}(F^e_*R)$ for all $e$, which ensures the positivity of the F-signature limit in strongly F-regular rings.