pith. sign in

arxiv: 2604.15592 · v1 · submitted 2026-04-17 · 🧮 math.AG

Algebraic Geometry over Non-Algebraically Closed Fields -- A-Coherent Sheaves over a Ringed Space

Hamet Seydi , Teylama Miabey This is my paper

Pith reviewed 2026-05-10 09:16 UTC · model grok-4.3

classification 🧮 math.AG
keywords A-coherent sheavesA-quasi-coherent sheavesringed spacesfinitely presented modulesequivalence of categoriesNash functionsanalytic functionsCartan's Theorem B
0
0 comments X

The pith

A-coherent sheaves on a ringed space are equivalent to finitely presented modules over the ring of global sections when the canonical morphism is flat and the global section functor is exact.

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

The paper defines an O_X-module to be A-coherent if it has a global presentation by free modules of finite rank over the ringed space X. It proves that this category is equivalent to the category of finitely presented modules over A equals Gamma of X and O_X, provided the canonical map from A to the structure sheaf is flat and taking global sections is exact. The authors apply the equivalence to establish faithful flatness for maps from rings of Nash functions to rings of analytic functions by invoking the vanishing of higher cohomology groups from Cartan's Theorem B.

Core claim

Under the assumptions that the canonical morphism from the ring A of global sections to the structure sheaf O_X is flat and that the global section functor is exact, the category of A-coherent O_X-modules is equivalent to the category of finitely presented A-modules. A-coherent means the sheaf admits a presentation by free O_X-modules of finite rank defined globally on X, and likewise A-quasi-coherent uses free modules of arbitrary rank.

What carries the argument

The A-coherent O_X-module, defined as an O_X-module that admits a global presentation by free modules of finite rank over the ringed space X.

Load-bearing premise

The canonical morphism from the ring of global sections to the structure sheaf must be flat and the global section functor must be exact.

What would settle it

Construct a ringed space X where the canonical morphism is flat and global sections are exact, yet there exists an A-coherent sheaf whose corresponding module is not finitely presented or the functor from sheaves to modules fails to be an equivalence.

read the original abstract

In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp. $A$-quasi-coherent) if it admits a global presentation by free modules of finite rank (resp. arbitrary rank) over a ringed space $X$. We establish a fundamental correspondence between these sheaves and modules over the ring of global sections $A = \Gamma(X,\mathcal{O}_X)$. Specifically, we prove that under conditions of flatness for the canonical morphism and the exactness of the global section functor, there exists an equivalence of categories between $A$-coherent $\mathcal{O}_X$-modules and finitely presented modules over $A$. We further demonstrate the utility of these results by proving the faithful flatness of the canonical homomorphisms from rings of Nash functions to rings of analytic functions, utilizing the vanishing of higher cohomology groups as guaranteed by Cartan's Theorem~B.

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

2 major / 2 minor

Summary. The paper defines A-coherent O_X-modules on a ringed space X (with A = Γ(X, O_X)) as those admitting a global finite free presentation by O_X-modules. It proves that, assuming flatness of the canonical morphism and exactness of the global sections functor, the functors Γ and sheafification induce an equivalence of categories between A-coherent O_X-modules and finitely presented A-modules (Theorem 3.4). The result is applied in §5 to prove faithful flatness of the canonical homomorphism from the ring of Nash functions to the ring of analytic functions, using Cartan's Theorem B to obtain vanishing of higher cohomology.

Significance. If the hypotheses hold in the target applications, the conditional equivalence offers a module-theoretic reduction for certain sheaf problems in algebraic geometry over non-algebraically closed fields, particularly in real settings. The explicit use of Cartan's Theorem B in the Nash-analytic application is a concrete strength, as it ties the abstract framework to a standard tool in analytic geometry and yields a verifiable prediction about flatness.

major comments (2)
  1. [Theorem 3.4 and §5] Theorem 3.4 assumes flatness of the canonical morphism as a hypothesis for the equivalence. Section 5 invokes this equivalence to establish faithful flatness of the Nash-to-analytic homomorphism. The manuscript must clarify whether flatness is verified independently for this specific morphism (e.g., via Cartan's Theorem B or another argument) or whether the application avoids using the equivalence in a way that presupposes the flatness being proved.
  2. [§5] In §5 the exactness of Γ is deduced from Cartan's Theorem B. It should be confirmed that this vanishing applies to the A-coherent sheaves in the equivalence (rather than only to standard coherent analytic sheaves on Stein spaces), since the A-coherent condition is defined via global presentations over A.
minor comments (2)
  1. [Abstract and §2] The abstract introduces both A-coherent and A-quasi-coherent sheaves, yet the main theorem and application address only the coherent case. A short remark in §2 or the introduction explaining why the equivalence does not immediately extend to the quasi-coherent setting would improve completeness.
  2. [§2 and §3] The notation for the associated-sheaf functor and the canonical morphism is used from §3 onward but is not defined until later; introducing these explicitly in §2 alongside the definitions of A-coherence would enhance readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit clarification on the logical dependencies in Theorem 3.4 and its application in §5. We have revised the manuscript to address both points directly, ensuring the argument is free of circularity and that Cartan's Theorem B is confirmed to apply to the relevant sheaves.

read point-by-point responses

  1. Referee: [Theorem 3.4 and §5] Theorem 3.4 assumes flatness of the canonical morphism as a hypothesis for the equivalence. Section 5 invokes this equivalence to establish faithful flatness of the Nash-to-analytic homomorphism. The manuscript must clarify whether flatness is verified independently for this specific morphism (e.g., via Cartan's Theorem B or another argument) or whether the application avoids using the equivalence in a way that presupposes the flatness being proved.

    Authors: We agree that the original presentation could be misread as potentially circular and have revised §5 to remove any ambiguity. Flatness of the canonical homomorphism φ: A_Nash → A_analytic is established independently of Theorem 3.4, using the fact that the Nash ring is a subring of the analytic ring and that φ is flat by the local flatness criterion for real analytic and Nash function rings (this is a standard fact in real analytic geometry, independent of our equivalence). With flatness in hand, we then apply the equivalence of Theorem 3.4 (whose hypotheses are now verified) together with the exactness of Γ (from Cartan's Theorem B) to conclude that φ is faithfully flat: if M ⊗_{A_Nash} A_analytic = 0 for a finitely presented A_Nash-module M, then the corresponding A-coherent sheaf vanishes, hence M = 0. A new paragraph at the start of §5 explicitly sequences the arguments and states that the equivalence is invoked only after flatness has been secured separately. revision: yes

  2. Referee: [§5] In §5 the exactness of Γ is deduced from Cartan's Theorem B. It should be confirmed that this vanishing applies to the A-coherent sheaves in the equivalence (rather than only to standard coherent analytic sheaves on Stein spaces), since the A-coherent condition is defined via global presentations over A.

    Authors: We have added a short lemma (now Lemma 5.1) confirming that every A-coherent O_X-module is coherent in the classical sense. By definition an A-coherent sheaf admits a global finite free presentation O_X^r → O_X^s → F → 0. On a Stein space the structure sheaf is coherent, finite free sheaves are coherent, and coherence is preserved under cokernels of morphisms between coherent sheaves. Consequently Cartan's Theorem B applies verbatim: H^i(X, F) = 0 for i > 0. The exactness of Γ on A-coherent sheaves therefore follows directly from the classical vanishing theorem rather than from any special property of the global A-presentation. The revised §5 cites this lemma before invoking the exactness of Γ in the proof of faithful flatness. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from definitions and external inputs

full rationale

The central result (Theorem 3.4) establishes a conditional equivalence between A-coherent sheaves (defined in §2 as O_X-modules admitting a global finite free presentation) and finitely presented A-modules, using only the stated flatness of the canonical morphism A → Γ(X, O_X) and exactness of the global sections functor to show that Γ and the associated-sheaf functor are inverses. This is a direct proof via preservation of exact sequences under flatness and recovery of modules under exactness of Γ; no step reduces by construction to a fitted input, renamed empirical pattern, or self-citation chain. Cartan's Theorem B is cited externally for vanishing in the §5 application to Nash-analytic maps and is not used to justify the equivalence itself. The manuscript is therefore self-contained against standard algebraic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claims rest on the newly introduced definitions of A-coherent and A-quasi-coherent sheaves together with standard domain assumptions from sheaf theory; no free parameters or invented entities beyond the definitions are required.

axioms (3)
  • domain assumption The canonical morphism from A to the stalks is flat
    Explicitly required for the category equivalence in the abstract.
  • domain assumption The global section functor is exact
    Explicitly required for the category equivalence in the abstract.
  • domain assumption Cartan's Theorem B applies to guarantee vanishing of higher cohomology
    Invoked for the faithful flatness application.
invented entities (1)
  • A-coherent sheaf no independent evidence
    purpose: To extend the notion of coherent sheaves to ringed spaces over non-algebraically closed fields via global finite-rank presentations
    Newly defined in the paper; no independent evidence outside the definition is supplied.

pith-pipeline@v0.9.0 · 5496 in / 1581 out tokens · 56687 ms · 2026-05-10T09:16:37.304783+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    Algebraic approximation of structures over complete local rings,

    M. Artin, “Algebraic approximation of structures over complete local rings,”Publ. Math. IH ´ES36(1969), 23–58

    work page 1969

  2. [2]

    Bochnak, M

    J. Bochnak, M. Coste, and M. F. Roy,Real Algebraic Geometry, Springer-Verlag, 1998

    work page 1998

  3. [3]

    Vari´ et´ es analytiques complexes et th´ eor` emes de finitude,

    H. Cartan, “Vari´ et´ es analytiques complexes et th´ eor` emes de finitude,”S´ eminaire Henri Cartan, 1951

    work page 1951

  4. [4]

    Sur l’anneau des fonctions de Nash,

    M. Raimondo, “Sur l’anneau des fonctions de Nash,”C. R. Acad. Sci. Paris290(1980)

    work page 1980

  5. [5]

    Faisceaux Alg´ ebriques Coh´ erents,

    J.-P. Serre, “Faisceaux Alg´ ebriques Coh´ erents,”Annals of Mathematics61(2) (1955), 197–278

    work page 1955

  6. [6]

    G´ eom´ etrie Alg´ ebrique et G´ eom´ etrie Analytique,

    J.-P. Serre, “G´ eom´ etrie Alg´ ebrique et G´ eom´ etrie Analytique,”Annales de l’Institut Fourier6(1956), 1–42

    work page 1956

  7. [7]

    Su una congettura di Nash,

    A. Tognoli, “Su una congettura di Nash,”Annali della Scuola Normale Superiore di Pisa27(1) (1973). Emeritus Professor, Department of Mathematics, Cheikh Anta Diop University, Dakar, Senegal. Email address:hsedi@gmail.com Department of Mathematics, University of the District of Columbia, W ashington DC, United States. Email address:teylama.miabey@udc.edu

    work page 1973