Embeddings and intersections of adelic groups

Dmitry Badulin

arxiv: 2510.22408 · v2 · submitted 2025-10-25 · 🧮 math.AG · math.AC

Embeddings and intersections of adelic groups

Dmitry Badulin This is my paper

Pith reviewed 2026-05-18 03:55 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords adelic groupsintersectionsprojective varietieslocally free sheavesexcellent schemesCohen-Macaulay schemesglobal sectionscurtailed complexes

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The pith

On three-dimensional regular projective varieties over countable fields, the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for locally free sheaves F.

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

The paper establishes embeddings of adelic groups on excellent schemes of special type equipped with flat quasicoherent sheaves. It proves specific intersection equalities for normal excellent schemes when index sets satisfy I ∩ J = I excluding zero. A limit result recovers global sections of locally free sheaves on Cohen-Macaulay projective schemes as limits of restrictions to power thickenings of integral subschemes. These tools yield an intersection theorem for adelic groups on normal projective surfaces and, via cohomology computations on curtailed adelic complexes, the main equality on three-dimensional regular projective varieties over countable fields. A sympathetic reader would care because the equality reduces the computation of intersections among adelic groups to a simple index operation, which may streamline work with global sections and cohomology in higher-dimensional geometry.

Core claim

We prove embeddings of adelic groups on an excellent scheme of special type and a flat quasicoherent sheaf on it. For a normal excellent scheme of special type we establish the equality A_I(X,F) ∩ A_J(X,F) = A_{I excluding 0}(X,F) in the case I ∩ J = I excluding 0. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes equals the group of global sections of this sheaf. Using this result, we deduce a theorem on intersections of adelic groups for normal projective surfaces. We also compute cohomology groups of a curtailed adelic complex and, as a consequence, show that on a three-dimend

What carries the argument

The adelic groups A_I(X,F) indexed by subsets I of {0,1,2,3}, which encode restrictions and completions of the sheaf F along subschemes of varying dimension on the variety X and carry the intersection and embedding arguments.

If this is right

Embeddings of adelic groups into larger adelic structures hold on excellent schemes of special type with flat quasicoherent sheaves.
The equality A_I ∩ A_J = A_{I excluding 0} holds whenever I ∩ J = I excluding 0 on normal excellent schemes of special type.
Global sections of locally free sheaves on Cohen-Macaulay projective schemes arise as limits of their restrictions to power thickenings of integral subschemes.
Cohomology groups of the curtailed adelic complex are determined by the same intersection and limit results.
The full intersection equality A_I ∩ A_J = A_{I∩J} extends to all pairs of index sets on three-dimensional regular projective varieties over countable fields.

Where Pith is reading between the lines

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

The countability assumption on the base field may be removable by replacing it with a finiteness or separability condition that still controls the relevant cohomology.
Similar intersection formulas could be tested on singular threefolds by relaxing regularity while retaining projectivity and the countable-field hypothesis.
The limit result on thickenings suggests that adelic groups may serve as a uniform replacement for direct limits in computations of sections on projective schemes of any dimension.

Load-bearing premise

The underlying scheme must be normal excellent of special type, or Cohen-Macaulay projective, or three-dimensional regular projective over a countable field.

What would settle it

An explicit computation or counterexample on a concrete three-dimensional regular projective variety over the rationals, such as projective three-space, where A_I(X,F) ∩ A_J(X,F) properly contains A_{I∩J}(X,F) for some choice of I, J and locally free F would disprove the equality.

read the original abstract

We prove embeddings of adelic groups on an excellent scheme of special type and a flat quasicoherent sheaf on it. For a normal excellent scheme of special type we establish the equality $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})=\mathbb{A}_{I\setminus0}(X,\mathcal{F})$ in the case $I\cap J=I\setminus0$. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes equals the group of global sections of this sheaf. Using this result, we deduce a theorem on intersections of adelic groups for normal projective surfaces. We also compute cohomology groups of a curtailed adelic complex and, as a consequence, show that on a three-dimensional regular projective variety over a countable field the intersection $\mathbb{A}_I(X,\mathcal{F})\cap\mathbb{A}_J(X,\mathcal{F})$ equals $\mathbb{A}_{I\cap J}(X,\mathcal{F})$ for any $I,J\subset\{0,1,2,3\}$ and any locally free sheaf $\mathcal{F}$ on $X$.

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 delivers concrete adelic intersection equalities on threefolds over countable fields built from supporting limit and embedding results.

read the letter

The one thing to know is that this paper gives a theorem on intersections of adelic groups on three-dimensional regular projective varieties over countable fields, specifically that the intersection of A_I and A_J is A_{I cap J} for any I and J in the power set of {0,1,2,3}. It also proves some preliminary results like embeddings of adelic groups on excellent schemes of special type with flat sheaves, the equality A_I cap A_J = A_{I minus 0} under the condition I cap J = I minus 0 for normal excellent schemes, and that the limit of global sections of locally free sheaves on power thickenings of subschemes equals the global sections on Cohen-Macaulay projective schemes. These are used to get the surface case and then the threefold result by computing cohomology of the curtailed adelic complex. The paper does well in organizing these steps into a coherent argument. The limit result on sections stands out as potentially reusable in other contexts involving thickenings. The soft spots are the restrictive hypotheses. The countable field and the special type or regular conditions limit how far the results apply, and it would help if the paper clarified the necessity of countability or gave more context on what special type means. The soundness looks okay from the chain described, with no obvious circularity or missing links in the logic. This paper is for specialists in algebraic geometry who use adelic methods for local-to-global questions, particularly those computing cohomology on projective varieties. A reader in that area could find the explicit intersection formulas useful for their own work. It has clear, specific claims that make it worth a serious referee's time. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves embeddings of adelic groups on excellent schemes of special type with flat quasicoherent sheaves. For normal excellent schemes of special type it establishes the equality A_I(X,F) ∩ A_J(X,F) = A_{I∖{0}}(X,F) when I ∩ J = I∖{0}. It shows that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes equals the group of global sections. These are used to deduce an intersection theorem for normal projective surfaces. The paper also computes cohomology groups of a curtailed adelic complex and, as a consequence, proves that on a three-dimensional regular projective variety over a countable field the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for any I,J ⊂ {0,1,2,3} and any locally free sheaf F.

Significance. If the central claims hold, the results extend the theory of adelic groups and their intersections to higher-dimensional schemes under hypotheses such as Cohen-Macaulay or regular projective varieties over countable fields. The limit-of-sections result and the cohomology computation of the curtailed complex provide concrete tools that could support further work on adelic cohomology and related structures in algebraic geometry.

major comments (2)

[Section on the limit result for global sections] The verification of the limit result (that lim Γ(X_n, F) = Γ(X, F) for locally free F on Cohen-Macaulay projective schemes) is presented as following from standard sheaf properties, but the manuscript does not supply explicit error bounds or a complete computation of the inverse limit; this step is load-bearing for the subsequent deduction of the intersection theorem on normal projective surfaces.
[Theorem on intersections for three-dimensional regular projective varieties] In the argument for the three-dimensional case, the equality A_I(X,F) ∩ A_J(X,F) = A_{I∩J}(X,F) is deduced by chaining the curtailed-complex cohomology computation with the restricted intersection equality on normal excellent schemes; however, the role of countability of the base field and the precise passage from the surface case to the threefold case are not fully detailed, leaving the robustness of the final claim difficult to assess without additional verification.

minor comments (2)

[Abstract] The notation I∖0 in the abstract and early statements should be written explicitly as I ∖ {0} to prevent misreading as a different operation.
[Introduction and statements of main theorems] Ensure that all references to 'special type' schemes include a precise definition or citation at first use, as the term appears without local clarification in the statements of the embedding and intersection results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major points below and will revise the manuscript to incorporate additional details and clarifications.

read point-by-point responses

Referee: [Section on the limit result for global sections] The verification of the limit result (that lim Γ(X_n, F) = Γ(X, F) for locally free F on Cohen-Macaulay projective schemes) is presented as following from standard sheaf properties, but the manuscript does not supply explicit error bounds or a complete computation of the inverse limit; this step is load-bearing for the subsequent deduction of the intersection theorem on normal projective surfaces.

Authors: We agree that a more explicit treatment of the inverse limit would improve clarity. The argument in the manuscript relies on the coherence of the sheaf and the fact that, for a Cohen-Macaulay projective scheme, the higher cohomology groups vanish in a manner that forces the restriction maps to stabilize. In the revised version we will add a self-contained computation showing that the inverse limit equals the global sections by verifying that the kernel and cokernel of the transition maps vanish for sufficiently large n, using the local cohomology exact sequence and the Cohen-Macaulay hypothesis to control the relevant Ext groups. revision: yes
Referee: [Theorem on intersections for three-dimensional regular projective varieties] In the argument for the three-dimensional case, the equality A_I(X,F) ∩ A_J(X,F) = A_{I∩J}(X,F) is deduced by chaining the curtailed-complex cohomology computation with the restricted intersection equality on normal excellent schemes; however, the role of countability of the base field and the precise passage from the surface case to the threefold case are not fully detailed, leaving the robustness of the final claim difficult to assess without additional verification.

Authors: The countability of the base field ensures that the relevant adelic groups and their intersections can be realized as countable unions, allowing us to apply the surface-case equality iteratively without introducing uncountable choices that would obstruct the identification. The passage from surfaces to threefolds proceeds by using the cohomology of the curtailed adelic complex to reduce the three-dimensional intersection to a collection of two-dimensional intersections on hyperplane sections. We will expand the relevant subsection to spell out this reduction step by step, including the precise invocation of countability in the direct-limit argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard results

full rationale

The paper derives embeddings of adelic groups on excellent schemes of special type, intersection equalities on normal excellent schemes, limit-of-sections results on Cohen-Macaulay projective schemes, and cohomology computations leading to the intersection theorem on three-dimensional regular projective varieties over countable fields. These steps rely on standard scheme-theoretic properties, sheaf cohomology, and explicit constructions without any reduction of a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The central claims for the three-dimensional case follow directly from the prior limit and equality results under the stated hypotheses, making the derivation independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard background properties of schemes and sheaves without introducing new free parameters or postulated entities.

axioms (3)

domain assumption Excellent schemes of special type admit the stated embeddings of adelic groups for flat quasicoherent sheaves.
Invoked directly in the first main theorem.
domain assumption For normal excellent schemes of special type the intersection equality holds when I ∩ J = I ∖ 0.
Central to the second theorem.
domain assumption On Cohen-Macaulay projective schemes the limit of restrictions of global sections to power thickenings equals the full global sections group.
Used to deduce the surface intersection theorem.

pith-pipeline@v0.9.0 · 5728 in / 1429 out tokens · 41085 ms · 2026-05-18T03:55:30.599282+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 echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

on a three-dimensional regular projective variety over a countable field the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for any I,J ⊂ {0,1,2,3}
IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

Theorem 3.2.2 ... AI(X,F)∩AJ(X,F)=AI∖0(X,F) for normal excellent scheme

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.