Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings

Andr\'e Dosea, Cleto B. Miranda-Neto, Majid Eghbali

Pith reviewed 2026-05-09 23:29 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords local cohomologyPeskine-Szpiro theoremcohomologically Mittag-Leffler ringssheaf cohomologythickeningsvanishing theoremsprime characteristicCohen-Macaulay ideals

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The pith

In cohomologically Mittag-Leffler rings of prime characteristic the cohomological dimension of any Cohen-Macaulay ideal equals its height and sheaf cohomology vanishes on thickenings.

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

The paper defines cohomologically Mittag-Leffler rings with respect to a flat local endomorphism such as the Frobenius. These rings include many classes with mild singularities such as Cohen-Macaulay rings, Stanley-Reisner rings, and Du Bois singularities. Using this definition the authors establish that certain maps on local cohomology modules are surjective, which yields a general version of the Peskine-Szpiro relation between the cohomological dimension and the height of a Cohen-Macaulay ideal. The same framework produces a Kodaira-type vanishing statement for the sheaf cohomology of infinitesimal thickenings of subschemes. A reader would care because the results move classical vanishing and dimension formulas from regular rings into a wider setting where explicit calculations are otherwise hard.

Core claim

In a cML ring the natural transition maps between local cohomology modules with support in an ideal I and with support in a power of I are epimorphisms. For any Cohen-Macaulay ideal I this epimorphism property implies that the cohomological dimension of I equals the height of I. The same epimorphism statements, read in the dual language of sheaf cohomology, imply that the higher cohomology groups of the structure sheaf on the nth infinitesimal thickening of a closed subscheme vanish.

What carries the argument

Cohomologically Mittag-Leffler rings with respect to a flat local endomorphism, which guarantee that the indicated transition maps on local cohomology modules are epimorphisms.

Load-bearing premise

The rings must be cohomologically Mittag-Leffler with respect to some flat local endomorphism in prime characteristic.

What would settle it

A concrete Cohen-Macaulay ideal I inside a cML ring for which the local cohomology module H_I to the power ht(I) plus one is nonzero would disprove the claimed equality of cohomological dimension and height.

read the original abstract

We study the surjectivity of certain maps involving local cohomology modules, which we can realize as a dual version of part of the investigation developed by Bhatt, Blickle, Lyubeznik, Singh and Zhang on the sheaf cohomology of thickenings (i.e., subschemes defined by powers of ideals), where injectivity played a central role. To this end, we introduce and investigate properties of cohomologically Mittag-Leffler (cML) rings, associated to a given flat local endomorphism (for instance the Frobenius map of a regular ring of prime characteristic), a class which we show to contain, in our setting, the so-called cohomologically full rings of Dao, De Stefani and Ma (in particular, Cohen-Macaulay, Stanley-Reisner, and Du Bois singularities) as well as rings with an ideal inducing a pure endomorphism of the quotient. Our two major specific goals rely upon the prime characteristic setting. First, we extend for the class of cML rings a classical result of Peskine and Szpiro that relates the cohomological dimension and the height of a given Cohen-Macaulay ideal. Second, we prove and illustrate a Kodaira type vanishing result on the sheaf cohomology of thickenings.

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 defines cohomologically Mittag-Leffler rings to extend Peskine-Szpiro and prove a Kodaira-type vanishing for thickenings in positive characteristic.

read the letter

Your colleague should know that this paper defines a class of cohomologically Mittag-Leffler rings relative to a flat local endomorphism and shows that it contains the cohomologically full rings from Dao, De Stefani, and Ma, along with some others. They then prove an extension of the Peskine-Szpiro theorem relating cohomological dimension to ideal height for Cohen-Macaulay ideals in these rings, and they give a Kodaira-type vanishing result for the sheaf cohomology of thickenings. The paper does a solid job connecting the surjectivity of maps on local cohomology to these classical results. It builds directly on the work of Bhatt, Blickle, Lyubeznik, Singh, and Zhang by looking at the dual picture with epimorphisms instead of injections. The examples and containments they check for the new class help make it feel grounded rather than artificial. The soft spots are mostly about scope. The results are tied to the prime characteristic setting and the existence of that flat endomorphism, so they won't apply in mixed characteristic or without such a map. The abstract presents the two theorems as major, but I wonder if the proofs require additional conditions on the rings or ideals that aren't highlighted upfront. Since the full details aren't in the abstract, the strength of the vanishing statement for thickenings needs checking against potential counterexamples or edge cases. This is aimed at commutative algebraists interested in local cohomology and positive characteristic techniques. Someone already familiar with Peskine-Szpiro and the thickenings literature will find the most value here, as it refines and extends those lines. It deserves to go to peer review. The new class and the stated theorems are substantive enough that referees should have a look to verify the derivations and see how far the applications reach.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces cohomologically Mittag-Leffler (cML) rings associated to a flat local endomorphism (e.g., the Frobenius in prime characteristic) and shows that this class contains cohomologically full rings, Cohen-Macaulay rings, Stanley-Reisner rings, and Du Bois singularities. It then extends the classical Peskine-Szpiro theorem by relating the cohomological dimension of a Cohen-Macaulay ideal to its height for cML rings, and proves a Kodaira-type vanishing theorem for the sheaf cohomology of thickenings, realized as a dual to recent injectivity results of Bhatt-Blickle-Lyubeznik-Singh-Zhang.

Significance. If the central claims hold, the work supplies a new, reasonably broad class of rings that unifies several important singularity classes and yields concrete generalizations of two classical results in local cohomology and vanishing theorems. The dual perspective on epimorphisms of local cohomology modules is a natural and potentially fruitful counterpart to existing injectivity studies; the explicit inclusion of multiple singularity types strengthens applicability.

minor comments (3)

The definition of cML rings (early in the manuscript) would benefit from an additional concrete example beyond the listed classes, to clarify how the Mittag-Leffler condition is verified in practice.
Notation for the flat local endomorphism and the associated local cohomology maps is introduced gradually; a single preliminary subsection collecting all standing notation would improve readability.
In the statement of the generalized Peskine-Szpiro result, the precise hypotheses on the ideal (Cohen-Macaulay and the cML condition) should be restated verbatim rather than referenced back to the definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No circularity detected in the derivation chain

full rationale

The paper defines the new class of cohomologically Mittag-Leffler (cML) rings via surjectivity properties of maps on local cohomology modules with respect to a given flat local endomorphism (e.g., Frobenius). It then verifies that this class contains standard examples such as Cohen-Macaulay rings, Stanley-Reisner rings, and Du Bois singularities by direct appeal to their known properties, and extends the classical Peskine-Szpiro theorem on cohomological dimension versus height for Cohen-Macaulay ideals to the cML setting. All steps rely on standard facts about local cohomology, flat endomorphisms, and prime-characteristic techniques; no result is obtained by fitting parameters to data and relabeling them as predictions, nor by load-bearing self-citations, self-definitional loops, or smuggling ansatzes. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of local cohomology modules, flat endomorphisms, and prime-characteristic Frobenius actions. No numerical free parameters are introduced. The cML rings are defined rather than postulated as new physical entities.

axioms (2)

standard math Local cohomology modules satisfy standard functorial properties and exact sequences under flat base change.
Invoked throughout the study of epimorphisms and the Peskine-Szpiro extension.
domain assumption In prime characteristic, the Frobenius endomorphism is flat on regular rings.
Used to associate cML rings to the Frobenius.

pith-pipeline@v0.9.0 · 5549 in / 1391 out tokens · 19983 ms · 2026-05-09T23:29:56.059011+00:00 · methodology

discussion (0)

Reference graph

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