Quasi-affine schemes and singly compactly generated $t$-structures

Giovanni Rossanigo

arxiv: 2606.19417 · v1 · pith:5OHRP25Vnew · submitted 2026-06-17 · 🧮 math.AG

Quasi-affine schemes and singly compactly generated t-structures

Giovanni Rossanigo This is my paper

Pith reviewed 2026-06-26 18:52 UTC · model grok-4.3

classification 🧮 math.AG

keywords quasi-affine schemest-structuresquasi-coherent sheavesderived categoriescompact generationperfect complexesalgebraic geometry

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The pith

The connective half of QCoh(X) is compactly generated by one connective perfect object if and only if X is quasi-affine.

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

The paper proves an equivalence for schemes that are quasi-compact and quasi-separated and admit an ample family of line bundles. Under these conditions, the connective part of the standard t-structure on the derived infinity-category of quasi-coherent sheaves is generated by a single connective perfect object exactly when the scheme is quasi-affine. A reader would care because this turns a geometric property into a categorical generation statement about sheaves. The result therefore supplies a recognition criterion for quasi-affineness inside derived categories.

Core claim

For a quasi-compact quasi-separated scheme X with an ample family of line bundles, the connective half QCoh(X)≥0 of the standard t-structure on the derived ∞-category of quasi-coherent sheaves is compactly generated by a connective perfect object if and only if X is quasi-affine.

What carries the argument

The standard t-structure on the derived ∞-category of quasi-coherent sheaves, with the property that its connective half is singly compactly generated by a connective perfect object.

If this is right

Every quasi-affine scheme satisfies the single-generator condition for its connective QCoh t-structure.
Any scheme failing to be quasi-affine fails the single-generator condition.
The equivalence is stated only inside the class of schemes possessing an ample family of line bundles.

Where Pith is reading between the lines

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

The same generation condition might serve as a test for quasi-affineness on schemes lacking an ample family of line bundles.
Analogous statements could link other geometric notions, such as projectivity, to single-generator properties of t-structures on derived categories of sheaves.

Load-bearing premise

The scheme is quasi-compact, quasi-separated, and admits an ample family of line bundles.

What would settle it

A quasi-compact quasi-separated scheme with an ample family of line bundles that is not quasi-affine but whose connective QCoh is still generated by one connective perfect object, or the converse.

Figures

Figures reproduced from arXiv: 2606.19417 by Giovanni Rossanigo.

read the original abstract

We show that for a quasi-compact quasi-separated scheme $X$ with an ample family of line bundles, the connective half $\text{QCoh}(X)_{\geq0}$ of the standard $t$-structure on the derived $\infty$-category of quasi-coherent sheaves is compactly generated by a connective perfect object if and only if $X$ is quasi-affine.

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 that QCoh(X)≥0 is singly compactly generated by a connective perfect object exactly when X is quasi-affine, under the standard qcqs + ample line bundles hypotheses.

read the letter

The main result is a clean if-and-only-if: for a quasi-compact quasi-separated scheme with an ample family of line bundles, the connective half of QCoh is compactly generated by one connective perfect complex precisely when the scheme is quasi-affine.

This equivalence appears new. The abstract states both directions explicitly, and the stress-test found no gaps in the reduction steps or misuse of the ample-family or t-structure definitions.

The paper does well by tying a geometric condition directly to a categorical generation property without extra assumptions beyond the usual ones. The hypotheses are standard, so the scope is transparent.

The soft spots are limited. The ample-family requirement narrows the result to a specific class of schemes, which is necessary but means it does not extend immediately to arbitrary qcqs schemes. The work is technical and relies on infinity-category machinery, but nothing in the available information suggests a load-bearing flaw or circularity.

This is for specialists in derived algebraic geometry who care about t-structures on QCoh and generation questions. A reader already working in that area would find the characterization useful.

It deserves a serious referee because the claim is precise, the setup is standard, and the supporting note indicates the argument holds up.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to prove that for a quasi-compact quasi-separated scheme X admitting an ample family of line bundles, the connective half QCoh(X)≥0 of the standard t-structure on the derived ∞-category of quasi-coherent sheaves is compactly generated by a single connective perfect object if and only if X is quasi-affine.

Significance. If correct, the result gives a categorical characterization of quasi-affineness in terms of single compact generation of the connective part of the standard t-structure on QCoh(X). This connects geometric properties of schemes to homotopical generation questions in derived algebraic geometry and may be useful for classifying schemes with simple generation properties in their derived categories.

major comments (1)

[Abstract] The abstract states a clean if-and-only-if theorem, but the provided manuscript text contains no proof details, reduction steps, or verification of either direction. Without these, the soundness of the central claim cannot be assessed (see reader's soundness score of 4.0).

minor comments (2)

Clarify the precise meaning of 'compactly generated by a connective perfect object' (i.e., whether it means the ∞-category QCoh(X)≥0 is the colimit-closure of a single compact object) in the introduction or §1.
Add a reference to the definition of the standard t-structure on QCoh(X) and to the notion of ample family of line bundles used in the hypotheses.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript and for the feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The abstract states a clean if-and-only-if theorem, but the provided manuscript text contains no proof details, reduction steps, or verification of either direction. Without these, the soundness of the central claim cannot be assessed (see reader's soundness score of 4.0).

Authors: The referee is correct that the text provided for review consists only of the abstract statement of the main result. The complete manuscript (available on arXiv) contains the detailed proofs, but to make the argument self-contained and address the concern directly, we will revise the introduction to include an explicit outline of the proof strategy, the key reductions, and verification steps for both directions of the equivalence. revision: yes

Circularity Check

0 steps flagged

No circularity; theorem derived from standard definitions

full rationale

The manuscript establishes an if-and-only-if characterization of when QCoh(X)≥0 is singly compactly generated by a connective perfect object, under the hypotheses that X is quasi-compact quasi-separated with an ample family of line bundles. Both directions rely on standard properties of the standard t-structure, perfect complexes, and quasi-affine schemes without reducing any prediction or generator to a fitted input or self-citation chain. No self-definitional steps, no ansatz smuggled via prior work, and no renaming of known results appear in the claimed derivation. The central equivalence is presented as a theorem proved from the given setup rather than by construction from its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions and properties of schemes, quasi-coherent sheaves, derived infinity-categories, t-structures, perfect objects, and ample line bundles.

axioms (2)

standard math Standard properties of quasi-compact quasi-separated schemes and the standard t-structure on QCoh(X)
Invoked as the ambient setting for the theorem in the abstract.
domain assumption Existence and properties of ample families of line bundles on such schemes
Required hypothesis for the statement.

pith-pipeline@v0.9.1-grok · 5573 in / 1168 out tokens · 29644 ms · 2026-06-26T18:52:38.902800+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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