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Classification and nonexistence for $t$-structures on derived categories of schemes

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Pith's one-line read Tensor t-structures on the bounded derived category of coherent sheaves are classified by support data for suitable Noetherian schemes, and their existence on perfect complexes detects regularity.

desk verdict Classification of tensor t-structures on coherent derived categories for suitable Noetherian schemes, plus an alternative proof that their restriction to perfect complexes detects regularity. read the letter →

arxiv 2404.08578 v5 pith:PBVWODTS submitted 2024-04-12 math.AG math.CT

Alexander Clark , Pat Lank , Kabeer Manali-Rahul , Chris J. Parker This is my paper · ORCID
classification math.AGmath.CT
keywords tensort-structuresderivedcategoriescoherentsheavesperfectcomplexesschemeregularityNoetherianschemessupportconditions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper classifies all tensor t-structures with prescribed support on the bounded derived category of coherent sheaves and related variants. It further proves that the existence of any such t-structure that restricts to the subcategory of perfect complexes is equivalent to the scheme being regular. The classification rests on local-to-global principles that reduce questions about global t-structures to local data. A reader would care because the result gives an explicit description of these structures and recovers a known detection theorem for regularity in the affine case by new methods.

What carries the argument

Tensor t-structures (t-structures compatible with the derived tensor product) on derived categories of coherent sheaves, classified via support conditions.

What would settle it

A concrete counterexample would be a suitable Noetherian scheme that is not regular yet admits a tensor t-structure on its perfect complexes.

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Extended reading notes

Core claim

Given a suitable Noetherian scheme X, the tensor t-structures on D^b(Coh(X)) with prescribed support are completely classified; moreover, the existence of a tensor t-structure on Perf(X) detects that X is regular, recovering Neeman's theorem in the affine case by different methods, while the same tools yield local-to-global principles for tensor t-structures.

Load-bearing premise

The scheme must be suitable Noetherian so that the support-based classification and local-to-global reduction apply.

Editorial extensions

If this is right

  • All tensor t-structures with given support on D^b(Coh(X)) can be listed explicitly from local data.
  • Regularity of X is equivalent to the existence of any tensor t-structure restricting to Perf(X).
  • Questions about global tensor t-structures reduce to local questions via the established principles.
  • The same classification applies to variants of the category with prescribed support.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The support classification may extend to unbounded derived categories if additional finiteness conditions are imposed.
  • The detection of regularity via t-structures could be tested on explicit non-affine examples such as projective space or singular curves.
  • Local-to-global principles might apply to other compatibility conditions beyond the tensor product.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript classifies tensor t-structures on the bounded derived category of coherent sheaves D^b_coh(X) (and variants with prescribed support) for a suitable Noetherian scheme X. It shows that the existence of such t-structures restricting to perfect complexes detects regularity of X, recovering Neeman's affine result by different methods, and establishes local-to-global principles for tensor t-structures.

Significance. If the classification and detection results hold, the work provides a useful extension of t-structure theory to coherent sheaves with support conditions and a new approach to regularity detection. The local-to-global principles are a potential strength for applications in derived algebraic geometry.

major comments (1)
  1. [Abstract and §1] Abstract and §1: the classification is stated for a 'suitable Noetherian scheme' but the precise conditions (quasi-compactness, separatedness, finite Krull dimension, etc.) that make the statements hold are not listed explicitly at the outset; this is load-bearing for the scope of the main theorems.
minor comments (1)
  1. Ensure that each main theorem statement includes a self-contained list of hypotheses on X rather than relying solely on the global 'suitable' qualifier.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the classification is stated for a 'suitable Noetherian scheme' but the precise conditions (quasi-compactness, separatedness, finite Krull dimension, etc.) that make the statements hold are not listed explicitly at the outset; this is load-bearing for the scope of the main theorems.

    Authors: We agree that the current phrasing leaves the precise hypotheses implicit at the outset. In the revised version we will explicitly list the standing assumptions (Noetherian, quasi-compact, separated, finite Krull dimension) both in the abstract and at the opening of §1, while retaining the detailed discussion already present in §2. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper classifies tensor t-structures on D^b_coh(X) and variants for suitable Noetherian schemes X, establishes local-to-global principles, and recovers Neeman's affine regularity result by different methods. The derivation relies on standard derived-category tools and external theorems rather than reducing any central claim to a self-citation chain, fitted parameter renamed as prediction, or self-definitional step. The 'suitable' scope conditions are explicit rather than hidden assumptions that force the result by construction. No load-bearing equation or uniqueness theorem is shown to collapse to the paper's own inputs.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No information on free parameters, axioms, or invented entities is extractable from the abstract alone.

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Cite this review

Pith. "Pith review of Classification and nonexistence for $t$-structures on derived categories of schemes." pith.science (2026). https://pith.science/paper/PBVWODTS

@misc{pith2026240408578,
  author       = {Pith},
  title        = {Pith review of: Classification and nonexistence for $t$-structures on derived categories of schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBVWODTS}},
  note         = {Machine review of arXiv:2404.08578}
}
abstract

Given a suitable Noetherian scheme, we classify tensor $t$-structures on the bounded derived category of coherent sheaves and its variants with prescribed support. Furthermore, we show that the existence of such $t$-structures restricting to perfect complexes detects regularity, recovering a theorem of Neeman in the affine case by different methods. Our tools establish local-to-global principles for tensor $t$-structures.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Semi-Bousfield classes and nonmonotone perversities

    math.CT 2026-05 unverdicted novelty 7.0 of 10

    Semi-Bousfield classes generalize Bousfield classes and tensor-compatible t-structures, with a bijection assigning (nonmonotone) perversities on Noetherian schemes X to semi-Bousfield classes in D_qc(X) that stratifie...

  2. Proxy smallness meets $t$-structures

    math.AG 2026-05 unverdicted novelty 6.5 of 10

    Defines proxy smallness for t-structures to characterize locally complete intersection schemes and classify preaisles on D^b_coh topologically.

Reference graph

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