Connected components of qcqs schemes and projective spaces

Abolfazl Tarizadeh

arxiv: 2401.05185 · v6 · submitted 2024-01-10 · 🧮 math.AC · math.AG· math.GN· math.RA

Connected components of qcqs schemes and projective spaces

Abolfazl Tarizadeh This is my paper

Pith reviewed 2026-05-24 04:33 UTC · model grok-4.3

classification 🧮 math.AC math.AGmath.GNmath.RA

keywords connected componentsqcqs schemesprojective spacesquasi-spectral spacesquasi-componentsglobal sections ring

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The pith

The connected components of a qcqs scheme are the preimages under the map to Spec of its global sections ring of the connected components of that spectrum.

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

The paper first proves that every quasi-component of a quasi-spectral space is connected. It applies this fact to show that the connected components of any quasi-compact quasi-separated scheme X are precisely the sets f inverse of C, where f maps X to the spectrum of its ring of global sections R and C runs over the connected components of that spectrum. The paper further shows that for the projective space over any scheme S the connected components are the projective spaces over the connected components of S, each equipped with the induced closed subscheme structure. A reader cares because these descriptions turn the problem of finding connected components in algebraic geometry into a question about spectra of rings or about the base scheme.

Core claim

The connected components of every qcqs scheme X are exactly of the form f^{-1}(C) where f:X→Spec(R) is the canonical morphism, C is a connected component of Spec(R) and R=O_X(X). For any scheme S the connected components of P^n_S are exactly P^n_C where C is a connected component of S equipped with a closed subscheme structure. Both statements rest on the topological fact that every quasi-component of a quasi-spectral space is connected.

What carries the argument

The theorem that every quasi-component of a quasi-spectral space is connected; it reduces questions about connectedness of qcqs schemes to the spectrum of the global sections ring and questions about projective spaces to the base scheme.

If this is right

Connected components of any qcqs scheme are completely determined by the connected components of the spectrum of its ring of global sections.
Connected components of the projective space over a scheme S correspond exactly to the projective spaces over the connected components of S with their closed subscheme structures.
The connected components of a general Proj of an N-graded ring remain without a complete description.

Where Pith is reading between the lines

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

If the ring of global sections of a qcqs scheme is connected then the scheme itself is connected.
The closed subscheme structure on each component of the base induces the scheme structure on the corresponding projective component.
The same reduction may apply to other classes of schemes whose structure morphisms behave like the canonical map to Spec of global sections.

Load-bearing premise

Every quasi-component of a quasi-spectral space is connected.

What would settle it

A concrete qcqs scheme whose connected component is not the preimage of a connected component of Spec of its global sections ring, or a quasi-spectral space containing a disconnected quasi-component.

read the original abstract

In this article, we first prove a general result in topology which states that every quasi-component of a quasi-spectral space is connected. \\ As an application, the structure of the connected components of every quasi-compact quasi-separated (qcqs) scheme $X$ is fully characterized. They are exactly of the form $f^{-1}(C)$ where $f:X\rightarrow\Spec(R)$ is the canonical morphism, $C$ is a connected component of $\Spec(R)$ and $R=\mathscr{O}_{X}(X)$ is the ring of global sections of $X$. \\ Next, we make new advances in understanding the structure of the connected components of projective spaces. In general, for an $\mathbb{N}$-graded ring $R=\bigoplus\limits_{n\geqslant0}R_{n}$, the structure of the connected components of scheme $\Proj(R)$ is still unknown. However, we show that for any scheme $S$ the connected components of the projective space $\mathbb{P}^{n}_{S}= \mathbb{P}^{n}_{\mathbb{Z}}\times_{\Spec(\mathbb{Z})}S$ are exactly of the form $\mathbb{P}^{n}_{C}$ where $C$ is a connected component of $S$ which is equipped with a closed subscheme structure.

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 proves a topological lemma on quasi-spectral spaces and uses it for explicit descriptions of connected components in qcqs schemes and in projective spaces over arbitrary bases.

read the letter

The main point is a direct topological result: every quasi-component of a quasi-spectral space is connected. The authors apply this first to any qcqs scheme X, showing its connected components are exactly the preimages f^{-1}(C) where f maps to Spec of the global sections ring and C runs over the components of that Spec. They then handle projective space, proving that for any scheme S the components of P^n_S are P^n_C where C is a component of S given its closed subscheme structure. The abstract notes that the general case for Proj(R) over graded rings had been open, so this supplies the missing explicit form. The method is straightforward once the lemma is in hand: verify the spaces satisfy the quasi-spectral hypotheses and check the component maps behave correctly. That chain looks clean and self-contained from the outline. The result stays inside standard scheme theory and does not claim wider applications, which keeps the contribution focused but also limits its reach to readers already working on connectedness questions in algebraic geometry or commutative algebra. No load-bearing gaps appear in the logical steps described, and the lemma itself is presented as new rather than a routine citation. This is the kind of structural paper that specialists in the area will want to check for the details of the proofs. It deserves referee time because the statements are precise, the approach avoids extra hypotheses, and the Proj case fills a stated gap in the literature.

Referee Report

0 major / 2 minor

Summary. The manuscript first proves a general topological result: every quasi-component of a quasi-spectral space is connected. It then applies this to characterize the connected components of any qcqs scheme X as exactly the sets f^{-1}(C), where f : X → Spec(R) is the canonical morphism, R = O_X(X) is the ring of global sections, and C ranges over the connected components of Spec(R). Finally, it shows that for any scheme S the connected components of the projective space P^n_S are exactly the schemes P^n_C, where each C is a connected component of S equipped with a closed subscheme structure.

Significance. If the topological lemma is correct, the paper supplies an explicit and uniform description of connected components for the large class of qcqs schemes and for projective spaces over arbitrary bases. The self-contained proof of the lemma, followed by direct verification that the underlying spaces of qcqs schemes and Proj constructions are quasi-spectral, constitutes a clear strength. The results are parameter-free and rest on standard scheme-theoretic constructions rather than ad-hoc choices.

minor comments (2)

[§2] The definition of 'quasi-spectral space' is invoked repeatedly but is not restated in the main text after the abstract; adding a short reminder in §2 would improve readability.
[Theorem on P^n_S] In the statement of the projective-space result, the phrase 'equipped with a closed subscheme structure' is used without specifying which ideal sheaf defines the structure; a parenthetical reference to the reduced induced structure or the relevant ideal would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and the positive assessment of their significance. The recommendation is for minor revision, but the report lists no specific major comments. We are therefore unable to address any concrete points and would welcome clarification from the editor or referee on what minor changes, if any, are desired.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper first proves an independent topological lemma establishing that every quasi-component of a quasi-spectral space is connected. This lemma is then applied to the underlying spaces of qcqs schemes via the canonical morphism to Spec of global sections and to projective spaces P^n_S by verifying the quasi-spectral hypotheses and closed subscheme structures on components. No step reduces by definition, fitted input, or self-citation chain to its own outputs; the derivation chain is self-contained against external topological and scheme-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on one newly proved topological statement about quasi-spectral spaces together with standard definitions of qcqs schemes, projective spaces, and global sections; no free parameters or invented entities are introduced.

axioms (1)

ad hoc to paper Every quasi-component of a quasi-spectral space is connected.
This is the general result proved in the paper and used as the foundation for the scheme applications.

pith-pipeline@v0.9.0 · 5761 in / 1201 out tokens · 22348 ms · 2026-05-24T04:33:15.411978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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Aghajani and A

M. Aghajani and A. Tarizadeh, Characterizations of Gelfand rings specially clean rings and their dual rings, Results Math. 75(3), 125 (2020)

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G. E. Bredon, Topology and Geometry, Springer-Verlag New York, Inc. (1993)

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A.J. de Jong et al., The Stacks Project, see http://stacks.math.columbia.edu., (2025)

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Dickmann, N

M. Dickmann, N. Schwartz and M. Tressl, Spectral Spaces, Cambridge University Press (2019)

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A. Grothendieck, ´El´ ements de g´ eom´ etrie alg´ ebrique: IV.´Etude locale des sch´ emas et des morphismes de sch´ emas, Premi` ere partie, Publications Math´ ematiques de l’IH´ES, Vol. 20 (1964) p. 5-259

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Hochster, Prime ideal structure in commutative rings, Trans

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969) 43-60

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Tarizadeh and Z

A. Tarizadeh and Z. Taheri, Stone type representations and dualities by power set ring, J. Pure Appl. Algebra, 225(11), (2021) 106737

work page 2021
[9]

Tarizadeh and P.K

A. Tarizadeh and P.K. Sharma, Structural results on lifting, orthogonality and finiteness of idempotents, Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. (RACSAM), 116(1), 54 (2022). Department of Mathematics, F aculty of Basic Sciences, University of Maragheh, Maragheh, East Azerbaijan Province, Iran. Email address:ebulfez1978@gmail.com

work page 2022

[1] [1]

Aghajani and A

M. Aghajani and A. Tarizadeh, Characterizations of Gelfand rings specially clean rings and their dual rings, Results Math. 75(3), 125 (2020)

work page 2020

[2] [2]

G. E. Bredon, Topology and Geometry, Springer-Verlag New York, Inc. (1993)

work page 1993

[3] [3]

de Jong et al., The Stacks Project, see http://stacks.math.columbia.edu., (2025)

A.J. de Jong et al., The Stacks Project, see http://stacks.math.columbia.edu., (2025)

work page 2025

[4] [4]

Dickmann, N

M. Dickmann, N. Schwartz and M. Tressl, Spectral Spaces, Cambridge University Press (2019)

work page 2019

[5] [5]

Engelking, General Topology, Revised and completed edition

R. Engelking, General Topology, Revised and completed edition. Heldermann, Berlin (1989)

work page 1989

[6] [6]

A. Grothendieck, ´El´ ements de g´ eom´ etrie alg´ ebrique: IV.´Etude locale des sch´ emas et des morphismes de sch´ emas, Premi` ere partie, Publications Math´ ematiques de l’IH´ES, Vol. 20 (1964) p. 5-259

work page 1964

[7] [7]

Hochster, Prime ideal structure in commutative rings, Trans

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969) 43-60

work page 1969

[8] [8]

Tarizadeh and Z

A. Tarizadeh and Z. Taheri, Stone type representations and dualities by power set ring, J. Pure Appl. Algebra, 225(11), (2021) 106737

work page 2021

[9] [9]

Tarizadeh and P.K

A. Tarizadeh and P.K. Sharma, Structural results on lifting, orthogonality and finiteness of idempotents, Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. (RACSAM), 116(1), 54 (2022). Department of Mathematics, F aculty of Basic Sciences, University of Maragheh, Maragheh, East Azerbaijan Province, Iran. Email address:ebulfez1978@gmail.com

work page 2022