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Proxy smallness meets $t$-structures

T0 review · 0 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read A scheme is locally a complete intersection exactly when every bounded coherent complex is t-⊗-proxy small.

desk verdict Solid upgrade of proxy-smallness characterizations of local complete intersections, plus usable classifications of ⊗-preaisles; the stalk-locality lemma holds up. read the letter →

arxiv 2605.26057 v3 pith:GTHAB22P submitted 2026-05-25 math.AG math.ACmath.CTmath.RT

Michal Hrbek , Pat Lank , Giovanna Le Gros , Sergio Pavon This is my paper · ORCID
classification math.AGmath.ACmath.CTmath.RT MSC 14A3014F0813D0918G8014B05
keywords proxysmallnesst-structurestensoractionslocalcompleteintersectionpreaislesThomasonfiltrationssingularitycategorypseudocoherentgeneration
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 refines the classical notion of proxy smallness by restricting the allowed operations to those of a t-structure (non-negative shifts, extensions, and direct summands) and by incorporating tensor actions of perfect complexes of non-positive amplitude. It proves that, for any Noetherian scheme, three conditions are equivalent: the scheme is locally a complete intersection; every object of the bounded derived category of coherent sheaves is ⊗-proxy small; and every such object is t-⊗-proxy small. Along the way the authors classify certain ⊗-preaisles by pairs consisting of a suspended subcategory of the singularity category and a Thomason filtration of the underlying space. The result upgrades earlier characterizations from the affine or separated setting to arbitrary Noetherian schemes and supplies an independent proof of recent classification theorems for hypersurface and complete-intersection singularities.

What carries the argument

t-⊗-proxy smallness: an object P is t-⊗-proxy small when the smallest cocomplete ⊗-preaisle it generates is compactly generated and the compact objects inside that preaisle already lie in the ordinary ⊗-preaisle generated by P. The notion is characterized by a gluing condition for the standard t-structure along Krause’s recollement.

What would settle it

Exhibit a Noetherian local ring that is not a complete intersection yet every object of its bounded coherent derived category is t-proxy small, or find a compactly generated aisle on K(Inj(X)) that fails to be stalk-local.

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Extended reading notes

Core claim

A Noetherian scheme X is locally a complete intersection if and only if every object of D^b_coh(X) is t-⊗-proxy small (equivalently, ⊗-proxy small). The same framework yields an injective map from the lattice of ⊗-suspended subcategories of D^b_coh(X) into the product of the lattice of ★-suspended subcategories of the singularity category and the set of Thomason filtrations on X; the map becomes bijective when X has only hypersurface singularities or arises as a zero locus of a section of a vector bundle on a regular scheme.

Load-bearing premise

Membership of an object in a compactly generated aisle of the homotopy category of injectives is completely determined by its stalks.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper introduces t-proxy smallness (and its tensor refinement t-⊗-proxy smallness) for objects and preaisles in triangulated categories associated to a Noetherian scheme X, working with tensor actions of Perf≤0(X) on Dqc(X), K(Inj(X)) and Sqc(X). The main structural result (Theorem 5.4 / Proposition 5.5) embeds the collection of t-⊗-proxy-small ⊗-suspended subcategories of Dbcoh(X) into pairs consisting of a ★-suspended subcategory of the singularity category and a Thomason filtration on X. As a geometric consequence, the authors prove that X is locally a complete intersection if and only if every object of Dbcoh(X) is ⊗-proxy small if and only if every object is t-⊗-proxy small (Proposition 1.2). When X has only hypersurface singularities, or arises as a zero locus of a section of a vector bundle on a regular scheme (Setup 7.6), the embedding becomes a bijection, yielding topological classifications of ⊗-suspended subcategories of Dbcoh(X) (Theorems 1.3 and 1.5). Intermediate technical results include the compact generation of ⊗-aisles generated by pseudocoherent complexes (Theorem 3.5) and a stalk-locality statement for membership in compactly generated ⊙-aisles of K(Inj(X)) (Lemma 6.8).

Significance. The work supplies a genuine t-structural refinement of the proxy-small characterizations of local complete intersections due to Pollitz and Briggs–Iyengar–Letz–Pollitz, and upgrades the latter from the separated to the arbitrary Noetherian setting. The stalk-locality lemma for aisles on K(Inj(X)) is a useful technical contribution that may find further applications. The classification theorems give a non-affine generalization of Takahashi’s recent results on suspended subcategories and are independent of his methods. The distinction between proxy smallness and t-proxy smallness is illustrated by concrete examples (Example 4.11), showing that the new notion is strictly finer. Overall the paper advances the interaction of tensor-triangular geometry, t-structures and singularity categories in a coherent and well-motivated way.

minor comments (4)
  1. [Introduction] In the introduction (p. 3) the phrase “Producing examples of t-proxy small objects of Dbcoh(X) which are not t-proxy small” is clearly a slip; the intended contrast is with ordinary proxy-small objects.
  2. [§2.4] Notation for the three tensor actions (⊗, ⊙, ★) is introduced only in Example 2.14; a brief forward reference earlier in §2.4 would help the reader.
  3. [Lemma 6.8] The proof of Lemma 6.8 invokes [Har13, III.6.8] for stalkwise vanishing of sheaf-Hom cohomology; a parenthetical reminder that the complexes are quasi-coherent would make the citation self-contained.
  4. [References] Several arXiv preprints are cited with temporary identifiers (e.g., HHLG26); once final versions appear they should be updated.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: new t-proxy notions and the LCI characterization rest on independent local-to-global arguments rather than definitional loops or load-bearing self-citations.

full rationale

The paper defines t-proxy smallness (Def. 4.2) and t-⊗-proxy smallness (Def. 5.1) by restricting classical proxy-small building (Def. 4.1, from DGI06/BIS25) to preaisle operations; these are not forced by prior notions and are shown strictly stronger via explicit counterexamples (Ex. 4.11, 4.17, 4.18). Proposition 1.2 equates local complete intersection with every object of D^b_coh(X) being ⊗-proxy small (recovering/upgrading BILP22) and t-⊗-proxy small. The (1)⇔(2) direction cites external results (Pol19, Let21, BILP22); the novel (1)⇒(3) direction reduces via the self-contained stalk-locality of compactly generated ⊙-aisles on K(Inj(X)) (Lemma 6.8, proved by reducing Hom-vanishing to stalkwise vanishing of sheaf-Hom cohomology and recovering global vanishing from compact generators) plus the affine local case (Prop. 6.9, using Bergh triangles and Thomason filtrations). Classifications (Thm. 5.4, 7.4, 7.7) embed into Stevenson’s external thick-subcategory classifications of D_sg plus Thomason filtrations (already bijective by ATJLS10/DS23/Lan25); the image criterion (Prop. 5.5) is an independent check, not a tautology. Self-citations (e.g., HHLG26, Lan25, BIL+26) supply background lemmas on aisles and actions that are independently established or machine-checkable in principle; none is the sole support for a uniqueness claim that forces the main equivalence. No fitted parameters, no ansatz smuggled as prediction, and no renaming of a known empirical pattern. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper works entirely inside the standard framework of triangulated categories of Noetherian schemes, Krause’s recollement, and Thomason filtrations. No free parameters are fitted. The only new entities are the definitions of t-proxy smallness and t-⊗-proxy smallness; everything else is either standard mathematics or a domain assumption already present in the literature the authors cite.

assumptions (4)
  • domain assumption Krause’s recollement relating K(Inj(X)), D_qc(X) and S_qc(X) exists for arbitrary Noetherian schemes (not merely separated ones).
    Invoked throughout §2.3 and used for all glueing arguments; justified by citing Kra05 + ČŠ20.
  • domain assumption Compactly generated tensor t-structures on D_qc(X) are classified by Thomason filtrations.
    Used in §2.5 and Theorem 3.5; taken from ATJLS10, DS23, Lan25.
  • domain assumption A Noetherian local ring is a complete intersection if and only if every object of D^b_coh(R) is proxy small.
    Pol19, Theorem 5.2; used as the local input for Proposition 1.2.
  • standard math Standard axioms of rigidly compactly generated tensor-triangulated categories and exact coproduct-preserving actions.
    Background for all tensor-action statements in §2.4.
invented entities (1)
  • t-proxy smallness (and its tensor variant t-⊗-proxy smallness)
    purpose: Refine classical proxy smallness so that only preaisle operations and a fixed tensor action are allowed; detect local complete intersections and classify preaisles.
    Definition 4.2 and Definition 5.1; the entire paper is built around these notions. No independent experimental handle exists; they are purely categorical.

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Pith. "Pith review of Proxy smallness meets $t$-structures." pith.science (2026). https://pith.science/paper/GTHAB22P

@misc{pith2026260526057,
  author       = {Pith},
  title        = {Pith review of: Proxy smallness meets $t$-structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTHAB22P}},
  note         = {Machine review of arXiv:2605.26057}
}
abstract

We introduce a notion of proxy smallness for $t$-structures on triangulated categories associated to a Noetherian scheme. Specifically, the theory is developed in the presence of tensor actions. Consequently, our results yield a new characterization of schemes that are locally complete intersections in terms of $t$-structures, as well as a topological classification of preaisles on the bounded derived category of coherent sheaves.

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Works this paper leans on

9 extracted references · 3 linked inside Pith

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Reviewed July 14, 2026 · model on record in the stance chip above.