A note concerning the vanishing of local cohomology for roots in mixed characteristic

Prashanth Sridhar

arxiv: 2506.09072 · v2 · submitted 2025-06-09 · 🧮 math.AC · math.AG

A note concerning the vanishing of local cohomology for roots in mixed characteristic

Prashanth Sridhar This is my paper

Pith reviewed 2026-05-19 10:26 UTC · model grok-4.3

classification 🧮 math.AC math.AG

keywords local cohomologyCohen-Macaulay ringsmixed characteristicintegral closureSerre condition

0 comments

The pith

For integral closures in p-th root extensions of mixed-characteristic regular rings, Cohen-Macaulayness holds exactly when the top local cohomology module vanishes.

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

The note considers an unramified regular local ring S of mixed characteristic p greater than zero and dimension d, together with the integral closure R of S inside a simple extension K obtained by adjoining a p-th root of an element of the fraction field L. It establishes that R is Cohen-Macaulay if and only if the local cohomology module H to the power d-1 with support at the maximal ideal of S vanishes. The same vanishing is shown to be equivalent to the dual module Hom from S to R satisfying Serre's condition S sub 3. A reader would care because this reduces the question of whether R satisfies the Cohen-Macaulay property to the behavior of a single cohomology module rather than a full set of depth conditions.

Core claim

Let (S, n) be an unramified regular local ring of mixed characteristic p > 0 and dimension d. Let L be the quotient field of S and K = L(ω) with ω^p belonging to L. Let R be the integral closure of S in K. Then R is Cohen-Macaulay if and only if H^{d-1}_n(R) = 0. Furthermore this vanishing is equivalent to the dual module Hom_S(R, S) satisfying Serre's condition (S_3).

What carries the argument

The integral closure R of S inside the extension K = L(ω), with the local cohomology module H^{d-1}_n(R) serving as the sole obstruction to the Cohen-Macaulay property of R.

If this is right

The obstruction to the Cohen-Macaulay property of R is concentrated in the single module H^{d-1}_n(R).
Vanishing of H^{d-1}_n(R) is equivalent to Hom_S(R, S) satisfying Serre's condition (S_3).
Cohen-Macaulayness of R can be tested by examining only this top local cohomology module rather than all depths.

Where Pith is reading between the lines

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

The result may simplify explicit checks of Cohen-Macaulayness for such rings by reducing the problem to a cohomology computation.
Similar reductions could be investigated for other ramified extensions or for rings that are not necessarily integral closures.

Load-bearing premise

S is an unramified regular local ring in mixed characteristic and R is the integral closure of S inside this specific extension by a p-th root.

What would settle it

Compute H^{d-1}_n(R) in a concrete low-dimensional example and check whether its vanishing coincides exactly with R being Cohen-Macaulay and with Hom_S(R, S) satisfying condition (S_3).

read the original abstract

The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(\omega)$ with $\omega^p\in L$. Let $R$ denote the integral closure of $S$ in $K$. Then $R$ is Cohen-Macaulay if and only if $\mathrm{H}^{d-1}_{\n}(R)=0$, i.e., the obstruction to the Cohen-Macaulayness of $R$ lies in a single local cohomology module. Furthermore, this is equivalent to the dual module $\Hom_S(R,S)$ satisfying Serre's condition $(S_3)$.

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

This note records a clean equivalence reducing Cohen-Macaulayness of a narrow class of mixed-characteristic integral closures to vanishing of one local cohomology module.

read the letter

This short note records that for an unramified regular local ring S of mixed characteristic and R its integral closure in the extension K = L(ω) with ω^p in L, R is Cohen-Macaulay if and only if H_n^{d-1}(R) vanishes. The same condition is equivalent to the dual Hom_S(R, S) satisfying Serre's (S_3). Local duality over S turns the claim into a statement about Ext_S^j(R, S) vanishing for j > 1, with lower cohomology modules vanishing automatically due to the extension degree and unramified setup. The paper does well by isolating this reduction cleanly and stating the equivalences without extra claims. The argument rests on standard tools and the specific hypotheses, with no evident internal contradictions or circular steps. The main soft spot is the very narrow scope: the result applies only to this controlled extension class and does not extend immediately to ramified cases or more general finite extensions. That keeps the note short and focused but limits its reach. This is the sort of precise observation that commutative algebraists working on local cohomology or arithmetic rings might keep on hand for reference. A reader already thinking about Cohen-Macaulay properties in mixed characteristic will get direct value from the stated equivalences. It deserves a serious referee because the claim is concrete, the setup is explicit, and the reduction is potentially useful even if modest in scope. I would send it out for peer review as a short note rather than desk reject.

Referee Report

1 major / 2 minor

Summary. The manuscript records a fact in mixed characteristic commutative algebra: Let (S, n) be an unramified regular local ring of mixed characteristic p>0 and dimension d. Let L be the fraction field of S and K = L(ω) with ω^p ∈ L. Let R be the integral closure of S in K. Then R is Cohen-Macaulay if and only if H_n^{d-1}(R) = 0 (i.e., the only possible obstruction to the Cohen-Macaulay property lies in this single local cohomology module). This is further equivalent to the dual module Hom_S(R, S) satisfying Serre's condition (S_3). The argument relies on automatic vanishing of H_n^i(R) for all i < d-1, converted via local duality over S into a statement about Ext_S^j(R, S) for j > 1.

Significance. If correct, the result gives a simplified, single-module criterion for Cohen-Macaulayness in this concrete class of rings obtained by adjoining a p-th root in mixed characteristic. The reduction via local duality to Ext vanishing is compatible with the finite degree dividing p and the unramified hypothesis on S, and may be useful for studying singularities or depth properties in this setting.

major comments (1)

[§2 (Main Theorem)] §2 (Main Theorem): The central claim that H_n^i(R) vanishes automatically for i < d-1 is load-bearing. The argument should explicitly isolate where the unramified hypothesis on S is used to control ramification in the extension K/L and ensure the vanishing does not require further conditions on the minimal polynomial of ω.

minor comments (2)

[Abstract and §1] Notation: the maximal ideal is written as n in the abstract and body but appears as 𝔫 in some displayed equations; standardize the symbol throughout.
[§3 (Proof of equivalence)] Add an explicit citation to the version of local duality (or Grothendieck duality) invoked when translating the local-cohomology statement into the Ext vanishing used for the (S_3) equivalence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our note, including the recommendation for minor revision. We address the major comment point by point below.

read point-by-point responses

Referee: The central claim that H_n^i(R) vanishes automatically for i < d-1 is load-bearing. The argument should explicitly isolate where the unramified hypothesis on S is used to control ramification in the extension K/L and ensure the vanishing does not require further conditions on the minimal polynomial of ω.

Authors: We agree that isolating the precise role of the unramified hypothesis would improve the clarity of the argument. In the proof of the main theorem, this hypothesis is used to guarantee that the finite extension K/L (of degree dividing p) remains tamely ramified with respect to the maximal ideal n, which in turn permits the application of standard vanishing results for local cohomology modules of finite modules over regular local rings in mixed characteristic; without it, additional conditions on the minimal polynomial of ω might be needed to control ramification. We will revise §2 to include a short clarifying remark that explicitly identifies this usage. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalences derived from local duality and standard vanishing results

full rationale

The paper records an equivalence: R is Cohen-Macaulay precisely when H^{d-1}_n(R) vanishes, with an equivalent formulation that Hom_S(R,S) satisfies (S3). This is obtained by applying local duality over the unramified regular ring S to convert the local-cohomology statement into one about Ext modules, combined with the finite extension degree dividing p and the unramified hypothesis to force automatic vanishing of H^i_n(R) for i < d-1. No step reduces a claimed prediction to a fitted input, renames a known pattern, or relies on a load-bearing self-citation whose justification is internal to the present work. The derivation therefore rests on externally verifiable commutative-algebra facts and does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions and properties from commutative algebra (local cohomology, Cohen-Macaulay rings, Serre conditions, integral closures) that are assumed known from prior literature. No free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)

standard math Local cohomology modules H_n^i(M) are well-defined and satisfy standard properties for modules over local rings.
Invoked directly when stating the vanishing condition for H^{d-1}_n(R).
standard math Serre's condition (S_k) is the standard depth condition on localizations of a module.
Used in the equivalence with Hom_S(R,S).

pith-pipeline@v0.9.0 · 5660 in / 1471 out tokens · 53023 ms · 2026-05-19T10:26:33.491836+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

R is Cohen-Macaulay if and only if H^{d-1}_n(R)=0, i.e., the obstruction to the Cohen-Macaulayness of R lies in a single local cohomology module. Furthermore, this is equivalent to the dual module Hom_S(R,S) satisfying Serre's condition (S3).

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.