Cartier algebras through the lens of $p$-families

Anna Brosowsky

arxiv: 2605.22987 · v1 · pith:Y2LQTPMXnew · submitted 2026-05-21 · 🧮 math.AC · math.AG

Cartier algebras through the lens of p-families

Anna Brosowsky This is my paper

Pith reviewed 2026-05-25 05:18 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords F-graded systemsCartier algebrasp-familiesstrong F-regularityF-splittingp-stabilizationmonomial ideals

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The pith

In a Gorenstein strongly F-regular local ring, a p-family is strongly F-regular exactly when its p-stabilization is F-split.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines F-graded systems of ideals whose sequences define Cartier algebras on a ring. It isolates p-families as a special class and proves that, inside local rings that are both Gorenstein and strongly F-regular, the two singularity measures of strong F-regularity and F-splitting become identical for these systems. A new operation called p-stabilization converts the question of whether the original system is strongly F-regular into the simpler question of whether the modified system is F-split. The same equivalence supplies an explicit criterion for the singularity properties of the Cartier algebra attached to the system. When the ideals are monomial, the paper also attaches a combinatorial object that computes the p-stabilization directly.

Core claim

In a Gorenstein and strongly F-regular local ring, strong F-regularity and F-splitting are the same for p-families; a system is strongly F-regular exactly when its p-stabilization is F-split. The paper uses this equivalence and the new p-stabilization operation to obtain a criterion for the singularity properties of the associated Cartier algebra, and it shows how a combinatorial object attached to monomial p-families computes the stabilization.

What carries the argument

p-stabilization, the operation on a p-family that produces a new system whose F-splitting status decides whether the original system is strongly F-regular.

If this is right

Strong F-regularity of any p-family reduces to checking F-splitting of one modified system.
The singularity type of the Cartier algebra is read off from the stabilized system.
For monomial p-families the stabilization is computed by a combinatorial object attached to the generators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The criterion may make it easier to test strong F-regularity in examples where F-splitting is already known or simpler to verify.
The combinatorial description for monomial cases could be turned into an explicit algorithm for deciding the property.
Similar stabilization constructions might be tried on other families of systems that generate Cartier algebras.

Load-bearing premise

The local ring must be both Gorenstein and strongly F-regular, and the ideal systems must be p-families.

What would settle it

A p-family inside a Gorenstein strongly F-regular local ring such that the system is strongly F-regular yet its p-stabilization fails to be F-split.

Figures

Figures reproduced from arXiv: 2605.22987 by Anna Brosowsky.

**Figure 1.** Figure 1: Plots of 1 p e log ae when I = ⟨x 3 , y6 ⟩, ae = Qe−1 i=0 I [p i ] , and p = 3. The relevant lattice points of 1 p e N lie above and to the right of the blue line in each subfigure [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

read the original abstract

We study $F$-graded systems of ideals in $R$, which are sequences of ideals giving rise to Cartier algebras on $R$. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the corresponding Cartier algebra. In particular, we show that in a Gorenstein and strongly $F$-regular local ring, strong $F$-regularity and $F$-splitting are the same for a special class of $F$-graded systems called $p$-families. Further, we make use of this and a new operation we introduce called $p$-stabilization to get a criterion that in a Gorenstein and strongly $F$-regular local ring, a system is strongly $F$-regular exactly when its $p$-stabilization is $F$-split. Finally, we associate a combinatorial object to systems built out of monomial ideals and show how this can help compute the $p$-stabilization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds p-stabilization and a monomial combinatorial tool for checking F-regularity of p-families, but only inside Gorenstein strongly F-regular rings.

read the letter

The main new items are the p-stabilization operation on p-families and the equivalence that strong F-regularity and F-splitting coincide for these systems in Gorenstein strongly F-regular local rings. They also give a criterion that reduces the check to whether the p-stabilization is F-split, plus a combinatorial object that helps compute the stabilization when the ideals are monomial. These pieces sit inside the existing theory of F-graded systems and Cartier algebras without obvious circularity. The conditions on the ring are stated clearly at the start, so the claims do not overreach their setting. The combinatorial construction looks like a practical addition for explicit calculations in the monomial case. The limitation is the narrow scope: the equivalence and criterion require the ring to already be Gorenstein and strongly F-regular, and the systems must be p-families. Outside that class the statements do not hold, which keeps the reach inside this subfield of F-singularities. The abstract presents the results as theorems, so the body needs to supply the arguments, but the setup itself shows no internal contradiction. This paper is for readers already working on Cartier algebras or test ideals in positive characteristic. Someone in that niche might use the criterion and the monomial tool for concrete examples. It deserves peer review because the operation and the combinatorial object are fresh within the area and rest on standard ring properties.

Referee Report

0 major / 2 minor

Summary. The manuscript studies F-graded systems of ideals in a ring R that give rise to Cartier algebras. It shows that in a Gorenstein and strongly F-regular local ring, strong F-regularity and F-splitting coincide for the special class of p-families. It introduces the p-stabilization operation and proves a criterion that a system is strongly F-regular exactly when its p-stabilization is F-split. It also associates a combinatorial object to systems built from monomial ideals to compute the p-stabilization.

Significance. If the stated equivalences and criterion hold, the work supplies new tools for relating singularity properties of Cartier algebras to properties of p-families, with a reduction of strong F-regularity checks to F-splitting of the stabilized system. The combinatorial construction for monomial systems offers a concrete computational aid in this setting.

minor comments (2)

Abstract: the main theorems are stated without cross-references to the sections containing their proofs or statements.
The definition and basic properties of p-families and the p-stabilization operation would benefit from an explicit low-dimensional example before the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our manuscript and for recommending minor revision. The report highlights the potential utility of our results relating strong F-regularity and F-splitting for p-families via p-stabilization, as well as the combinatorial tools for monomial cases. No specific major comments or requested changes are listed in the report.

Circularity Check

0 steps flagged

No significant circularity; claims rest on standard definitions

full rationale

The paper states theorems establishing equivalence of strong F-regularity and F-splitting for p-families, plus a p-stabilization criterion, but only under the explicit hypotheses that the ring is Gorenstein and strongly F-regular (local) and the systems are p-families. These are standard notions in the literature on Cartier algebras and F-singularities; the abstract and described structure give no indication that any load-bearing step reduces by definition, by fitting a parameter to the target quantity, or by a self-citation chain whose cited result itself presupposes the present claim. The restriction to this setting is presented as necessary rather than smuggled in. No equations or operations are described that would make a 'prediction' tautological with its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract alone gives no explicit free parameters or invented entities; results rest on standard domain assumptions about Gorenstein and strongly F-regular rings in positive characteristic.

axioms (1)

domain assumption The ring is Gorenstein and strongly F-regular local ring in positive characteristic
All main statements are conditioned on this property of the ring.

pith-pipeline@v0.9.0 · 5685 in / 1166 out tokens · 26801 ms · 2026-05-25T05:18:09.434250+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A (Theorem 4.3). Let (R,m) be a Gorenstein and strongly F-regular F-finite local ring. Let b• be a p-family in R. Then b• is F-split if and only if it is strongly F-regular.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 5.1. The p-stabilization of a• is ea• where ea_e = {r | r^{p^f} ∈ a_{e+f} for all f ≫ 0}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Test ideals via algebras ofp −e-linear maps

[Bli13] Manuel Blickle. “Test ideals via algebras ofp −e-linear maps”. In:Journal of Algebraic Geometry 22.1 (2013), pp. 49–83.issn: 1056-3911.doi:10.1090/S1056-3911-2012-00576-1(cit. on p. 5). [Bro23] Anna Brosowsky. “The Cartier Core Map for Cartier Algebras”. In:Journal of Algebra630 (Sept. 2023), pp. 274–296.issn: 0021-8693.doi:10.1016/j.jalgebra.2023...

work page doi:10.1090/s1056-3911-2012-00576-1(cit 2013
[2]

Uniform Bounds and Symbolic Powers on Smooth Varieties

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.isbn: 978-0-8218- 7287-1.doi:10.1090/gsm/130(cit. on p. 7). [ELS01] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith. “Uniform Bounds and Symbolic Powers on Smooth Varieties”. In:Inventiones Mathematicae144.2 (2001), pp. 241–252.issn: 0020-9910,1432- 1297.doi:10.1007/s...

work page doi:10.1090/gsm/130(cit 2012
[3]

Local Okounkov Bodies and Limits in Prime Character- istic

1989, pp. 119–133 (cit. on pp. 8, 9). [HJ18] Daniel J. Hern´ andez and Jack Jeffries. “Local Okounkov Bodies and Limits in Prime Character- istic”. In:Mathematische Annalen372.1-2 (2018), pp. 139–178.issn: 0025-5831.doi:10.1007/ s00208-018-1651-6(cit. on pp. 1, 6, 16, 20). [HL02] Craig Huneke and Graham J. Leuschke. “Two Theorems about Maximal Cohen-Macau...

work page 1989
[4]

Characterizations of regular local rings of characteristicp

London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006.isbn: 978-0-521-68860-4 (cit. on p. 7). [Kun69] Ernst Kunz. “Characterizations of regular local rings of characteristicp”. In:American Journal of Mathematics91 (1969), pp. 772–784.issn: 0002-9327.doi:10.2307/2373351(cit. on p. 3). [Kun76] Ernst Kunz. “On Noetheria...

work page doi:10.2307/2373351(cit 2006

[1] [1]

Test ideals via algebras ofp −e-linear maps

[Bli13] Manuel Blickle. “Test ideals via algebras ofp −e-linear maps”. In:Journal of Algebraic Geometry 22.1 (2013), pp. 49–83.issn: 1056-3911.doi:10.1090/S1056-3911-2012-00576-1(cit. on p. 5). [Bro23] Anna Brosowsky. “The Cartier Core Map for Cartier Algebras”. In:Journal of Algebra630 (Sept. 2023), pp. 274–296.issn: 0021-8693.doi:10.1016/j.jalgebra.2023...

work page doi:10.1090/s1056-3911-2012-00576-1(cit 2013

[2] [2]

Uniform Bounds and Symbolic Powers on Smooth Varieties

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.isbn: 978-0-8218- 7287-1.doi:10.1090/gsm/130(cit. on p. 7). [ELS01] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith. “Uniform Bounds and Symbolic Powers on Smooth Varieties”. In:Inventiones Mathematicae144.2 (2001), pp. 241–252.issn: 0020-9910,1432- 1297.doi:10.1007/s...

work page doi:10.1090/gsm/130(cit 2012

[3] [3]

Local Okounkov Bodies and Limits in Prime Character- istic

1989, pp. 119–133 (cit. on pp. 8, 9). [HJ18] Daniel J. Hern´ andez and Jack Jeffries. “Local Okounkov Bodies and Limits in Prime Character- istic”. In:Mathematische Annalen372.1-2 (2018), pp. 139–178.issn: 0025-5831.doi:10.1007/ s00208-018-1651-6(cit. on pp. 1, 6, 16, 20). [HL02] Craig Huneke and Graham J. Leuschke. “Two Theorems about Maximal Cohen-Macau...

work page 1989

[4] [4]

Characterizations of regular local rings of characteristicp

London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006.isbn: 978-0-521-68860-4 (cit. on p. 7). [Kun69] Ernst Kunz. “Characterizations of regular local rings of characteristicp”. In:American Journal of Mathematics91 (1969), pp. 772–784.issn: 0002-9327.doi:10.2307/2373351(cit. on p. 3). [Kun76] Ernst Kunz. “On Noetheria...

work page doi:10.2307/2373351(cit 2006