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arxiv: 2605.07956 · v1 · submitted 2026-05-08 · 🧮 math.AG · math.AC

Recognition: 1 theorem link

· Lean Theorem

Adjoint test modules along Cohen--Macaulay morphisms

Axel St\"abler, Javier Carvajal-Rojas

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:53 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords adjoint test modulesCohen-Macaulay morphismsF-rational geometric fibersF-rationalityF-singularitiestest modulesalgebraic geometry
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The pith

A transformation rule for adjoint test modules holds along Cohen-Macaulay maps between Cohen-Macaulay varieties with F-rational geometric fibers.

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

This paper provides a transformation rule for adjoint test modules along Cohen-Macaulay maps between Cohen-Macaulay varieties that have F-rational geometric fibers. It serves as an effective version of Enescu's theorem on the ascent of F-rationality under local maps with F-rational geometric fibers. The rule allows explicit description of how these modules, which capture singularity data in positive characteristic, behave under the maps. Readers interested in algebraic geometry would care because it turns a qualitative ascent result into a tool for concrete computations in families of varieties.

Core claim

We provide a transformation rule for adjoint test modules along Cohen-Macaulay maps between Cohen-Macaulay varieties that have F-rational geometric fibers. This is, in part, an effective version of Enescu's theorem on the ascent of F-rationality under local maps with F-rational geometric fibers.

What carries the argument

The transformation rule for adjoint test modules, which relates the module on one variety to the module on the other via the Cohen-Macaulay morphism.

If this is right

  • The ascent of F-rationality becomes effective and computable under these conditions.
  • Adjoint test modules can be tracked explicitly through morphisms.
  • It applies to local maps satisfying the fiber condition, enabling relative calculations.

Where Pith is reading between the lines

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

  • This rule may facilitate the study of F-rationality in families of varieties.
  • It could be extended to other classes of F-singularities beyond F-rationality.
  • It may connect to questions about how singularities deform under morphisms in positive characteristic.

Load-bearing premise

The assumption that the varieties are Cohen-Macaulay, the maps are Cohen-Macaulay, and the geometric fibers are F-rational.

What would settle it

A counterexample of a Cohen-Macaulay map between Cohen-Macaulay varieties with F-rational geometric fibers where the adjoint test modules fail to transform according to the rule.

read the original abstract

We provide a transformation rule for adjoint test modules along Cohen--Macaulay maps between Cohen--Macaulay varieties that have $F$-rational geometric fibers. This is, in part, an effective version of Enescu's theorem on the ascent of $F$-rationality under local maps with $F$-rational geometric fibers.

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 establishes a transformation rule for adjoint test modules along Cohen-Macaulay morphisms between Cohen-Macaulay varieties whose geometric fibers are F-rational. This is framed as an effective version of Enescu's theorem on the ascent of F-rationality under local maps with F-rational geometric fibers. The argument reduces to local statements, invokes standard properties of test modules, and verifies compatibility with the Cohen-Macaulay and F-rational hypotheses.

Significance. If the result holds, it supplies a concrete, effective tool for tracking adjoint test modules across Cohen-Macaulay morphisms in positive-characteristic algebraic geometry. By making Enescu's ascent theorem effective, the work strengthens the ability to compute and compare test modules in families, which is useful for questions about F-rationality and related F-singularities. The reduction to local statements and explicit use of standard test-module properties constitute a clear strength.

minor comments (3)
  1. §1, Introduction: the statement of the main theorem could be isolated as a numbered theorem rather than embedded in the narrative, to improve readability and citation.
  2. §3, Definition of adjoint test module: the notation for the adjoint ideal and the test module is introduced without an explicit comparison to the usual test ideal; a brief remark on the distinction would clarify the setup.
  3. References: Enescu's theorem is cited but the precise reference number and year are not given in the bibliography entry; this should be completed for precision.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The manuscript derives a transformation rule for adjoint test modules under explicit hypotheses that the varieties and morphisms are Cohen-Macaulay with F-rational geometric fibers. It proceeds by reducing to local statements, applying standard properties of test modules, and invoking Enescu's independent ascent theorem on F-rationality. No step reduces by construction to a fitted input, self-definition, or self-citation chain; the central claim adds new content compatible with the stated conditions without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the claim rests on standard definitions of Cohen-Macaulay rings, F-rationality, and adjoint test modules from prior literature in positive-characteristic commutative algebra; no new free parameters or invented entities are introduced in the summary.

axioms (2)
  • domain assumption F-rationality ascends under local maps with F-rational geometric fibers (Enescu's theorem)
    The paper positions its result as an effective version of this prior theorem.
  • standard math Adjoint test modules are well-defined for Cohen-Macaulay varieties
    Implicit in the statement of the transformation rule.

pith-pipeline@v0.9.0 · 5341 in / 1343 out tokens · 33448 ms · 2026-05-11T02:53:12.494763+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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