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arxiv: 2604.18568 · v2 · submitted 2026-04-20 · 🧮 math.AG · math.AC

Pulling back Cartier structures along regular maps

Javier Carvajal-Rojas , Axel St\"abler This is my paper

Pith reviewed 2026-05-10 03:31 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords Cartier modulesrelative Cartier isomorphismF-finite morphismsregular morphismsmixed test idealstest idealsNoetherian schemes
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The pith

Regular F-finite maps admit a relative Cartier isomorphism for pulling back modules.

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

The paper establishes a framework to pull back Cartier modules along regular F-finite morphisms of locally Noetherian schemes. It does so by constructing a relative Cartier isomorphism and a relative Cartier operator. This construction applies to arbitrary such maps. As a result, new statements about the regions where mixed test ideals remain constant follow from earlier work. A reader would care because it provides a general tool for moving invariants around in positive characteristic algebraic geometry without restricting to special classes of maps.

Core claim

We construct a relative Cartier isomorphism and operator for an arbitrary regular F-finite map of locally noetherian schemes. This allows pulling back Cartier modules and their associated invariants along regular F-finite morphisms, yielding new results on the constancy regions of mixed test ideals.

What carries the argument

The relative Cartier isomorphism and relative Cartier operator, which generalize the standard Cartier structure from the Frobenius to general regular F-finite maps.

If this is right

  • New results follow on the constancy regions of mixed test ideals.
  • Cartier modules and their invariants can be pulled back along any regular F-finite map.
  • The framework extends previous constructions to arbitrary such morphisms.
  • Associated invariants are transferred while maintaining their key properties.

Where Pith is reading between the lines

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

  • The pullback may help analyze how test ideals behave in flat families of varieties.
  • Similar techniques could extend to other invariants in positive characteristic geometry.
  • Verification on specific examples like regular maps of projective spaces would test the operator.

Load-bearing premise

The maps are regular and F-finite between locally Noetherian schemes.

What would settle it

A specific regular F-finite morphism of locally Noetherian schemes where the relative Cartier isomorphism fails to exist or satisfy the required properties would falsify the main result.

Figures

Figures reproduced from arXiv: 2604.18568 by Axel St\"abler, Javier Carvajal-Rojas.

Figure 1
Figure 1. Figure 1: A recursive way to obtain the boundary for p = 3. The graph on the left is scaled by 1/3 and replaces the graph in the intervals [0, 1/3] and [2/3, 1]. The so obtained graph is then scaled by 1/3 and replaces the graph in the intervals [0, 1/3] and [2/3, 1] in the final picture. Iterating this process indefinitely gives the boundary region. for all e ≫ 0. Since the ideal inside κ e is principal and homogen… view at source ↗
read the original abstract

We introduce a framework for pulling back Cartier modules and their associated invariants along regular $F$-finite morphisms. To achieve this, we construct a relative Cartier isomorphism and operator for an arbitrary regular $F$-finite map of locally noetherian schemes. As an application, we obtain new results on the constancy regions of mixed test ideals, based on the work of Felipe P\'erez.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs a relative Cartier isomorphism and associated pullback operator for Cartier modules along arbitrary regular F-finite morphisms of locally Noetherian schemes. This framework is then applied to derive new results on the constancy regions of mixed test ideals, building on prior work of Pérez.

Significance. If the construction is valid, the result supplies a functorial pullback for Cartier structures under the stated hypotheses, which are the natural conditions guaranteeing local freeness of the relative Frobenius. This extends the absolute theory in a way that directly supports computations of test ideals and related invariants in relative settings, potentially enabling new statements about constancy loci and base-change behavior.

minor comments (3)
  1. [§2] §2: The definition of the relative Cartier operator (around the displayed isomorphism following the statement of the main theorem) would benefit from an explicit verification that the construction is independent of the choice of local coordinates used to define the relative Frobenius.
  2. [§4] §4 (application to mixed test ideals): The statement of the new constancy-region result should include a precise comparison with the corresponding statement in Pérez's work, indicating exactly which hypotheses are relaxed.
  3. The manuscript would be improved by adding a short remark on whether the pullback commutes with the formation of test ideals in the expected way (i.e., whether the diagram relating the pulled-back Cartier module to the test ideal commutes).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report accurately captures the main results: the construction of a relative Cartier isomorphism and pullback operator for Cartier modules along arbitrary regular F-finite morphisms of locally Noetherian schemes, together with the application to constancy regions of mixed test ideals extending work of Pérez. Since the referee report lists no specific major comments, we have no individual points requiring point-by-point response or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; direct construction under stated hypotheses

full rationale

The paper's central contribution is the explicit construction of a relative Cartier isomorphism and associated pullback operator for regular F-finite morphisms of locally Noetherian schemes. This is presented as a direct, functorial construction relying on local freeness of the relative Frobenius and flatness, without any reduction of the output to fitted parameters, self-definitional loops, or load-bearing self-citations. The application to constancy regions of mixed test ideals is explicitly based on external prior work by Pérez, preserving independence. No equation or step in the derivation chain is shown to be equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided summary.

pith-pipeline@v0.9.0 · 5344 in / 846 out tokens · 25972 ms · 2026-05-10T03:31:52.138133+00:00 · methodology

discussion (0)

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Reference graph

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