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Convexity in tensor triangular geometry

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Pith's one-line read Convex subsets classify dualizable localizing ideals in tt-geometry

desk verdict Clean generalization of Efimov's dualizable ideal classification, but its hard direction leans on an unproved external theorem that a referee must check. read the letter →

arxiv 2506.12429 v2 pith:VJRP5CSH submitted 2025-06-14 math.CT math.AGmath.AT

Changhan Zou This is my paper · ORCID
classification math.CTmath.AGmath.AT MSC 18G8018N60
keywords tensortriangulargeometryBalmerspectrumlocalizingidealsdualizablecategoriesconvexsubsetscohomologicalstratificationLurieproductderivedofschemes
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 establishes a classification theorem for a large class of tensor triangular categories: rigidly-compactly generated tensor triangular $\infty$-categories that are locally cohomologically stratified and have noetherian Balmer spectrum. In such a category, the dualizable localizing ideals---those with a well-behaved dual under the Lurie tensor product---are in inclusion-preserving bijection with the convex subsets of the Balmer spectrum. The bijection sends an ideal to its tensor triangular support, and the inverse sends a convex subset to the localizing ideal of objects supported there. If the paper is right, this fills the middle rung of a three-level hierarchy of ideal classifications, between all localizing ideals (arbitrary subsets) and compactly generated localizing ideals (specialization closed subsets), and extends a known affine classification to settings such as derived categories of noetherian schemes.

What carries the argument

The argument turns on four interacting pieces. Local cohomological stratification lets the paper compute supports through local cohomology of the endomorphism ring. A quoted dualizable version of the Neeman-Thomason localization theorem, stated as Theorem 3.14, supplies the compact object needed to detect a contradiction in the local case. The characterization of convex subsets as differences $S_1 \setminus S_2$ of specialization closed subsets connects convexity to finite localizations. The tensor product of presentable stable categories, whose dualizability is stable under base change and localization by Lemmas 2.12 and 2.13, transports the nonconvexity obstruction from a local category back to the original one. Tensor triangular support itself, the set of primes where an object remains nonzero after tensoring with the idempotent $g_P$, is the map that carries the bijection.

What would settle it

Construct a rigidly-compactly generated tt-$\infty$-category that is locally cohomologically stratified with noetherian Balmer spectrum, and exhibit a dualizable localizing ideal whose support is not convex; Theorem 4.9 asserts none exists. Equivalently, find a convex subset $S$ of such a spectrum for which $\mathcal{C}_S$ is not dualizable. A concrete place to look is the local Lemma 4.21: a subset containing the unique closed point and a point $Q$ with $\overline{\{Q\}} \nsubseteq S$ but $S \cap (\overline{\{Q\}}\setminus\{Q\})$ specialization closed should force $\mathcal{C}_{S\cap\overline{\{Q\}}}$ to be non-dualizable, so producing a category where it is dualizable would refute the claim.

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Extended reading notes

Core claim

The central claim is Theorem 4.9. It asserts that for a rigidly-compactly generated tt-$\infty$-category $\mathcal{C}$ that is locally cohomologically stratified and has noetherian $\mathrm{Spc}(\mathcal{C}^c)$, tensor triangular support induces an inclusion-preserving bijection from dualizable localizing ideals of $\mathcal{C}$ to convex subsets of $\mathrm{Spc}(\mathcal{C}^c)$, with inverse $S \mapsto \mathcal{C}_S$. The theorem also asserts that each dualizable localizing ideal is itself compactly generated as a stable $\infty$-category. The proof first reduces to the case where the category is stratified, then treats a local category with a unique closed point: there, Lemma 4.21 shows that an ideal supported on a nonconvex 'punctured closure' slice cannot be dualizable, and global nonconvexity is pushed forward through finite localizations and relative tensor products to produce a contradiction.

Load-bearing premise

The load-bearing premise is a quoted theorem, not proved here, saying that localizations of dualizable stable categories behave like the classical Neeman-Thomason localization theorem; if that statement fails at the stated level of generality, the argument that a nonconvex support forces a non-dualizable ideal breaks.

Editorial extensions

If this is right

  • For every noetherian scheme $X$, the dualizable localizing ideals of $\mathcal{D}_{\mathrm{qc}}(X)$ correspond to the convex subsets of $X$.
  • Every dualizable localizing ideal in the classified categories is compactly generated, refining the classical description of compactly generated ideals in terms of specialization closed subsets.
  • The affine classification for commutative noetherian rings is recovered as the special case $\mathcal{C} = \mathcal{D}(R)$.
  • Cohomologically stratified categories, including many examples from modular representation theory and equivariant homotopy theory, fall under the theorem via Corollary 4.24.
  • The inclusion-preserving bijection means convexity is exactly the finiteness property that dualizability imposes on localizing ideals in this setting.

Reading between the lines

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

  • Editorial inference: the theorem suggests that dualizability is the categorical finiteness condition that turns arbitrary localizing-ideal classifications into convex-set classifications, so convexity could play a similar role in any stratified tensor triangular category with a closed tensor product.
  • A testable extension would be to drop local cohomological stratification and ask whether plain stratification plus a noetherian spectrum already forces the classification; the proof leans on support computations from local cohomology, so that hypothesis is the first one to probe.
  • The paper's Remark 4.23 indicates the dualizable-convex bijection can survive in some non-noetherian settings; identifying precisely which non-noetherian rings satisfy it would sharpen the boundary of the theorem.
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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

2 major / 6 minor

Summary. The paper studies rigidly-compactly generated tensor triangulated ∞-categories and their dualizable localizing ideals. The main theorem (Theorem 4.9) asserts that, under local cohomological stratification and a noetherian Balmer spectrum, tensor triangular support induces an inclusion-preserving bijection between dualizable localizing ideals and convex subsets of Spc(Cc), and that such ideals are compactly generated. The proof proceeds by reducing the hard direction to a local lemma (Lemma 4.21), using Efimov's extension of the Neeman–Thomason theorem, BIK support theory, and support computations from the author's earlier work. The paper also applies the result to derived categories of noetherian schemes, generalizing Efimov's affine classification.

Significance. If correct, the paper gives a clean structural classification in tensor triangular geometry that unifies Neeman-type and Efimov-type results at a high level of generality. The proof is coherent and does not assume the conclusion: it uses stratification, local-to-global principles, and support identification as inputs, and the author explicitly records in Remark 4.22 that without Efimov's dualizable Neeman–Thomason theorem only the easier direction is available. The main caveat is that the hard direction depends on Theorem 3.14, which is quoted from an unpublished preprint and not proved in the manuscript.

major comments (2)
  1. [§3.14, Lemma 4.21] The hard direction of Theorem 4.9 depends on Theorem 3.14, Efimov's extension of the Neeman–Thomason localization theorem to dualizable categories. The theorem is quoted from the unpublished preprint [Efi24] and no proof is supplied. It is stronger than the classical Neeman–Thomason theorem because it asserts q(a) ≅ x ⊕ Σx rather than only that x is a direct summand of q(a), and it is precisely what produces the compact object t in Lemma 4.21. The authors should prove this theorem in the paper, or give a precise, publicly verifiable reference with all hypotheses checked; as Remark 4.22 acknowledges, without it the converse implication in Theorem 4.9 is unavailable.
  2. [§3.18, §3.23, Theorem 4.9 proof] Several support-theoretic inputs are taken from the author's unpublished preprint [Zou23]: Theorem 9.3 (identification of tt-support with BIK support), Example 6.1 (localization at a prime), and Corollary 5.30 (support under base change). These statements are used at load-bearing points in Lemma 4.18, Proposition 3.23, and the proof of Theorem 4.9. The paper should state these results with precise hypotheses or give proofs for the cases used, so that the reader can verify the translation between BIK support and tensor triangular support.
minor comments (6)
  1. [§3.12] The word 'spetrum' in Example 3.12 should be 'spectrum'.
  2. [§4.12] In the proof of Lemma 4.12, 'To prove thatα is an isomorphism' is missing a space; it should read 'To prove that α is an isomorphism'.
  3. [§4.25] In Example 4.25, 'stratifed' should be 'stratified'.
  4. [§4.22] In Remark 4.22, 'the the extension' contains a duplicated article and should be corrected.
  5. [Equation (3.7)] The notation T^⊥ in equation (3.7) is used without being defined; please add a sentence defining it as the right orthogonal of T_{S2} in T.
  6. [§4.12] The Brown representability step in Lemma 4.12 would be easier to check if the author briefly noted that the functor Hom_R(H^*_1(-), I) sends coproducts to products because the unit is compact.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 4.9 is a genuine reduction to external localization theorems and standard stratification results, not to its own conclusion.

full rationale

The central claim (Theorem 4.9) is the classification of dualizable localizing ideals by convex subsets of the Balmer spectrum. This statement is not assumed anywhere. The easy direction, convex S implies C_S is dualizable, is proved directly via Proposition 4.7 using Lemma 3.11, compact generation of a quotient, and [BHV18, Lemma 2.17]. The hard direction, dualizable C_S implies S is convex, is a genuine reduction: the proof localizes at a prime, obtains a contradiction through Lemma 4.21, and uses the dualizable Neeman–Thomason theorem quoted from Efimov (Theorem 3.14) to produce a compact object t. That theorem is an external dependency on Efimov's preprint, not a restatement of the target classification, and the paper's own Remark 4.22 makes the dependence explicit: without Theorem 3.14, only the weaker compactly-generated correspondence is obtained. The author's previous work [Zou23] is cited only for support-theoretic identification results (Theorem 9.3 and Corollary 5.30), namely the equality of Balmer support and BIK support, and the behaviour of support under base change. These are used as tools in Lemmas 4.18 and 4.20 and in the proof of Theorem 4.9; they do not assert or imply the dualizable-ideal/convex-subset bijection. No parameter is fitted and no quantity is renamed as a prediction. The proof therefore does not reduce by construction to its own inputs, and the self-citations are conventional reliance on the author's earlier independent support-theory work rather than a circular load-bearing chain.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities are present. The central claim rests on standard higher categorical machinery plus several quoted stratification and support-theory theorems, two of which come from the author's previous preprint [Zou23]. The proof itself contributes a new combination of these inputs rather than introducing new objects or fitting parameters.

assumptions (7)
  • domain assumption Efimov's dualizable Neeman-Thomason localization theorem (Theorem 3.14) is correct as stated.
    Invoked in Lemma 4.21 to produce a compact object t of C_{S1} with q(t) isomorphic to y plus Sigma y; not proved in the paper and cited from the preprint [Efi24].
  • domain assumption The local-to-global principle and minimality criterion for stratification from [BHS23, Theorem 3.22 and Corollary 5.3] hold in the stated generality.
    Used in Proposition 3.23 to prove that local cohomological stratification plus noetherian spectrum implies stratification.
  • standard math The comparison map from the Balmer spectrum to the homogeneous Zariski spectrum of the endomorphism ring is a homeomorphism under cohomological stratification, with the localization identifications used in Proposition 3.23.
    Relies on [Bal10] and [Lau23], plus [Zou23, Example 6.1] to identify the localization at a prime with algebraic localization.
  • domain assumption The Benson-Iyengar-Krause support computations in [BIK08] are valid, including the specialization closure of support and the support formulas in local categories.
    Used in Lemma 4.18 and Corollary 4.20 to show that the relevant supports are specialization closed and to identify objects supported at the closed point.
  • domain assumption The author's earlier support-theoretic identifications in [Zou23, Theorem 9.3 and Corollary 5.30] are correct.
    Self-cited: these provide the equality of Balmer support and BIK support, and the behavior of support under finite localization. They are preprint results not independently verified here.
  • standard math Background on locally rigid and rigid infty-categories from [Ram24] and Lurie's Higher Algebra is correct, including the behavior of relative tensor products and dualizability.
    Definitions and lemmas in Section 2, including preservation of fully faithful morphisms by relative tensor product in Lemma 2.12, rely on these references.
  • standard math Brown representability and injective cogeneration over noetherian graded local rings are available in the triangulated setting.
    Used in Lemma 4.12 and Lemma 4.18 to construct representing objects and infinite direct sums of injectives; standard results but not proved in the paper.

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Pith. "Pith review of Convexity in tensor triangular geometry." pith.science (2026). https://pith.science/paper/VJRP5CSH

@misc{pith2026250612429,
  author       = {Pith},
  title        = {Pith review of: Convexity in tensor triangular geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJRP5CSH}},
  note         = {Machine review of arXiv:2506.12429}
}
abstract

We classify the dualizable localizing ideals of rigidly-compactly generated tt-$\infty$-categories that are cohomologically stratified. By definition, these are the localizing ideals that are dualizable with respect to the Lurie tensor product. We prove that these ideals correspond to the convex subsets of the Balmer spectrum. More generally, we establish this classification for categories which are locally cohomologically stratified and whose Balmer spectrum is noetherian. The classification thus applies to many categories arising in algebra and topology, including derived categories of noetherian schemes. Our result generalizes, and is motivated by, a recent theorem of Efimov which establishes this classification for derived categories of commutative noetherian rings.

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