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REVIEW 3 major objections 4 minor 23 references

Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems

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Pith's one-line read The paper establishes a duality between the compact side of a triangulated category and a non-compact subcategory built from Brown–Comenetz duals, and proves two representability theorems that identify this dual side with homological functo

desk verdict A genuinely dual representability program with a plausible central theorem, but Theorem 5.18 has a real proof gap at F∈[E]_4 and Theorem 6.5 assumes t-exactness without proof; deserves revision and a referee, not rejection. read the letter →

arxiv 2602.14383 v2 pith:FF5WEFUC submitted 2026-02-16 math.RT math.RA

Xiaohu Chen , Yongliang Sun , Yaohua Zhang This is my paper · ORCID
classification math.RTmath.RA MSC 18G8016E35
keywords triangulatedcategoriescompactsiltingobjectBrown–ComenetzdualityE-homologicalfunctorsrepresentabilityrecollementlocalizationderivedofalgebras
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 tries to build a mirror image of the compact-object side of a triangulated category, using Brown–Comenetz duality instead of compact generators. It constructs a subcategory T_c^+ on the non-compact side and characterizes it purely homologically: an object lies in T_c^+ exactly when its Hom functors into the Brown–Comenetz dual category are locally finite and vanish in high degrees. Under the compact silting hypothesis, the paper proves that every locally finite E-homological functor is represented by an object of T_c^+, and every finite E-homological functor by an object of the bounded subcategory T_c^b. This matters because it gives a concrete, generator-free description of which functors come from actual objects, and it yields localization short exact sequences on the injective/non-compact side of recollements. A sympathetic reader would see this as completing a duality between compact and non-compact localization phenomena in triangulated categories.

What carries the argument

Brown–Comenetz duality is treated as a partial Serre functor S:T_c→T: for a compact object G, Hom_T(−,S(G)) ≃ D Hom_T(G,−), assigning to each compact object a non-compact dual object. The category E is the thick subcategory generated by these duals of compact objects. The second load-bearing tool is a strong E-coapproximating system: a tower ⋯→E_3→E_2→E_1 inside E whose cohomology stabilizes in all low degrees. Homotopy limits of such towers are exactly the objects of T_c^+, and this characterization is what transports finiteness and vanishing properties from E to T_c^+. The Yoneda functor restricted to E is then the bridge connecting objects of T_c^+ to locally finite and finite E-homologic

What would settle it

Compute T_c^+ for a finite-dimensional algebra A, for instance A = k[x]/(x^2), and use the locally finite E-homological functor H = Hom_T(−, E). The theorem predicts H is represented by an object of T_c^+; check via Proposition 4.6 that the representing object is a homotopy limit of a strong E-coapproximating system. Any mismatch refutes the representability claim. On the localization side, inspect a recollement of finite-dimensional algebras and test whether i_* sends R^{≥0} into some T^{≥n}; a failure of this t-exactness would disprove the unstated premise behind Theorem 6.5(2).

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Brown–Comenetz dual category E plays the role of an injective-side analogue of the compact subcategory, and the intrinsic subcategory T_c^+ is exactly the homological shadow of E. For a locally Hom-finite k-linear triangulated category with a compact silting object G, Theorem 5.18 states that the Yoneda functor from (T_c^+)^op to Hom_k(E,k-Mod) is full, with essential image precisely the locally finite E-homological functors. Theorem 5.22 states that the restricted functor from (T_c^b)^op is fully faithful, with essential image precisely the finite E-homological functors. Before these representability results, Proposition 4.6 charac

Load-bearing premise

The proof of the localization theorem for T_c^+/T_b^c assumes, with no supporting argument, that the recollement functor i_* is t-exact for the preferred t-structures, so that i_*(R^{≥0}) lies in T^{≥n} for some integer n; if this premise fails, the induced maps on Verdier quotients need not be exact and the short exact sequences in Theorem 6.5(2) can break.

Editorial extensions

If this is right

  • Every locally finite E-homological functor is represented by an object of T_c^+, and every finite E-homological functor by an object of T_c^b; on the bounded side, the representation is unique up to isomorphism.
  • Membership in T_c^+ and T_c^b becomes intrinsic and homological: an object belongs to T_c^+ (resp. T_c^b) exactly when Hom into E is locally finite (resp. finite) and vanishes in the appropriate degrees.
  • For recollements of derived categories of finite-dimensional algebras, the Brown–Comenetz side restricts to a short exact sequence K^b(C-inj) → K^b(A-inj) → K^b(B-inj), complementing the compact-projective sequence.
  • The localization theorems produce short exact sequences of Verdier quotients such as S_c^+/E_s → T_c^+/E_t → R_c^+/E_r and S_c^+/S_c → T_c^+/T_c → R_c^+/R_c, giving a dual picture of how recollements decompose the non-compact side.
  • A triangle functor out of T_c^b has a left adjoint exactly when three explicit Hom-finiteness and vanishing conditions hold, giving a dual analogue of existing right-adjoint criteria.

Reading between the lines

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

  • Editorial extension: because the description of T_c^+ is homological and depends only on E and finiteness of Hom spaces, the same representability framework may identify T_c^+ in any locally Hom-finite triangulated category where E cogenerates, without needing an explicit silting generator.
  • Editorial extension: the finite-E-homological classification suggests a derived-equivalence test: two silting objects with equivalent categories of finite E-homological functors should be derived equivalent; one could attempt to prove this by comparing their Yoneda images.
  • Editorial extension: the localization results point toward dual ladder/adjoint constructions on the injective side; a natural test would be to use the three conditions in Proposition 7.1 to construct left adjoints for functors from singularity categories modeled by T_c^b.
  • Editorial extension: the strong-E-coapproximating-system characterization may give an algorithm for deciding membership in T_c^+ in concrete examples, namely by constructing cohomology-stabilized towers in E and checking whether their homotopy limits match the given object.
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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

3 major / 4 minor

Summary. The paper develops a Brown–Comenetz dual framework for Neeman's representability theorems in triangulated categories with a single compact generator / compact silting object. It defines the thick subcategory E generated by Brown–Comenetz duals of compact objects and the intrinsic subcategory T_c^+, gives a characterization of T_c^+ via strong E-coapproximating systems and homotopy limits, and proves two representability theorems: the Yoneda functor from (T_c^+)^op to Hom_k(E, k-Mod) is full with essential image the locally finite E-homological functors, and its restriction to (T_c^b)^op is fully faithful with essential image the finite E-homological functors. The paper also proves localization/recollement theorems for T_c^+ and E, and applies the results to derived categories of finite-dimensional algebras and to the construction of adjoints.

Significance. If the main theorems are correct, the paper gives a substantial dual counterpart to Neeman's theory of T_c^- and provides new structural information about the non-compact side of triangulated categories with compact silting objects. The use of Brown–Comenetz duality to construct an injective-side analogue of the compact subcategory is natural and potentially useful, and the paper includes a fair amount of supporting material: explicit coapproximating systems, homotopy-limit characterizations, recollement restrictions, and examples for derived categories of algebras. The main theorems are attractive and would be of interest to researchers in representation theory and triangulated category theory. However, the proof of the central representability theorem contains a load-bearing gap, and one localization theorem depends on an unproved t-exactness assertion; these issues must be resolved before the claims can be regarded as established.

major comments (3)
  1. [Theorem 5.18, proof (Section 5.3, line after Proposition 5.7)] The proof asserts 'Moreover, by the proof of Lemma 5.5 together with Proposition 5.7, we have F∈[E]_4.' This membership is essential: Lemma 5.17 is applied with C=F to conclude Hom_T(g,F)=0, and this vanishing is what forces F to be a direct summand of F_4 and hence an object of T_c^+. But no derivation of F∈[E]_4 is given. Lemma 5.5 only constructs objects F_i admitting strong <E>_i-coapproximating systems, which is a weaker statement than belonging to [E]_i. In the base case F_1 is an infinite coproduct of shifted copies of E, whereas [E]_1 is defined replacing coproducts by products (Section 2.1), so the distinction is non-trivial. The proof of Theorem 5.18 therefore does not establish that every locally finite E-homological functor is represented by an object of T_c^+, and this gap propagates to Theorem 5.22 and Corollary 5.19/5.23.
  2. [Theorem 6.5(2), proof (Section 6.1)] The proof states 'Because i_* is t-exact' without proof or reference. In a recollement, each category carries its own preferred t-structure determined by its compact silting object, and t-exactness of i_* is not automatic. The factorization argument for a morphism X→i_*(Y) uses the inclusion i_*(R^{≥0})⊆T^{≥n} for some integer n; this is exactly the t-exactness being assumed. If it fails, Claim 1 and Claim 2 need not hold, and the induced short exact sequence of Verdier quotients is not established. The authors should either prove this t-exactness from the recollement hypotheses or state and prove a suitable lemma; alternatively they should add an explicit hypothesis.
  3. [Lemma 5.4 and its uses (Sections 5.1–5.2)] Lemma 5.4 is described as a 'specialization' of [15, Lemma 8.5], but it is used in a dual setting involving homotopy limits, E-coapproximating systems, and Brown–Comenetz duality, whereas the cited lemma was proved in the compact-approximation/homotopy-colimit setting. The paper does not prove Lemma 5.4, nor does it verify that all hypotheses of [15, Lemma 8.5] transfer to the present context. This matters because Lemma 5.4 is used repeatedly in the inductive constructions of Lemmas 5.5, 5.12, and 5.13, which are in turn needed for Proposition 5.14 and Theorem 5.18. A complete proof or a precise statement of the dual version is required.
minor comments (4)
  1. [Section 2, Definition 2.1 (after preferred t-structures)] The displayed definition 'T_c^b := T_c^b ∩ T^b' is self-referential and clearly a typo; it should presumably be 'T_c^b := T_c^- ∩ T^b' (or similar).
  2. [Example 5.8(2)] The example states that Ho(Sp), the stable homotopy category of spectra, is a locally Hom-finite k-linear triangulated category with compact silting object the sphere spectrum. This is not standard: Ho(Sp) is not naturally k-linear over a field in the sense used elsewhere in the paper. The example needs clarification or deletion.
  3. [Throughout] There are numerous typos and minor language issues, e.g. 'approxiamble' in the abstract, 'sequneces' in Section 2.3, and inconsistent notation for the partial Serre functors (S_s, S_t, S_r) in Proposition 3.3. These should be corrected.
  4. [Section 5, Definition 5.15] The term 'weak triangle' is used in Lemma 5.4 and elsewhere but is only referenced to [15, Definition 8.2]; for the paper to be self-contained, the definition should be recalled or the reference made explicit at first use.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central representability theorems are derived from Neeman's results and Brown–Comenetz duality, not from the paper's own conclusions.

full rationale

The central representability results (Theorems 5.18 and 5.22) are not circular. The paper defines E as the Brown–Comenetz dual of the compact generator and T_c^+ independently via triangles with E-terms and high t-degree; it then proves, using Neeman's approximable triangulated category machinery ([15, Thm 9.18, 9.20], [15, Lemmas 7.8, 8.4–8.6]) and internal lemmas (4.5–5.17), that the Yoneda functor from (T_c^+)^op has as essential image exactly the locally finite E-homological functors. This is a substantive bijective statement, not an equality forced by definition. The only self-citation is [23] (Sun–Zhang), used in Section 6 for the localization theorems (Lemma 6.3, Theorem 6.5); it is not load-bearing for the central representability theorems and therefore does not make the derivation circular. Two genuine proof gaps appear — the unproved membership F∈[E]_4 in the proof of Theorem 5.18 ('Moreover, by the proof of Lemma 5.5 together with Proposition 5.7, we have F∈[E]_4') and the unstated t-exactness of i_* in Theorem 6.5(2) ('Because i_* is t-exact') — but these are correctness/completeness concerns, not instances of a conclusion being equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claims rest on standard Brown representability and partial Serre duality, plus a large body of prior results by Neeman and by the authors' own [23]. The new subcategory T_c^+ is the main invented object; the paper gives it two independent characterizations. No numerical free parameters are fitted.

assumptions (6)
  • standard math Brown representability theorem for compactly generated triangulated categories
    Used in Section 3 to define the Brown–Comenetz dual E of a compact object G.
  • domain assumption Partial Serre duality results of Oppermann–Psaroudakis–Stai [21, Theorem 3.3, Observation 3.4]
    Gives the triangle functor S:T^c→T with Hom(-,S(G))≃D Hom(G,-); local Hom-finiteness yields S fully faithful (Lemma 3.1).
  • domain assumption Neeman's approximable triangulated category results [15, Theorem 9.18, 9.20, Lemmas 7.8, 8.4, 8.5, 8.6]
    The representability and localization proofs invoke these lemmas as black boxes.
  • domain assumption Sun–Zhang localization theorems [23, Lemma 2.10, Corollary 3.3]
    Used in the proof of Theorem 6.5 and Lemma 6.3; note these are by two of the present authors.
  • domain assumption Preferred t-structure is nondegenerate and T≤0 is closed under products
    Required by Lemmas 2.4, 2.5, 4.5–4.7; for a compact silting object, T≤0 = G^{⊥>0} is closed under products, but nondegeneracy is imported from [8, Lemma 3.6].
  • standard math Bondal–Van den Bergh: T^c is Karoubian [5, Proposition 2.1.1]
    Used in Lemma 3.1(2) to show Im(S) is thick.
invented entities (1)
  • T_c^+ (intrinsic subcategory) independent evidence
    purpose: Formal dual of Neeman's T_c^-; objects approximated from below by Brown–Comenetz duals of compact objects.
    Defined intrinsically in Section 4 and characterized by homotopy limits of strong E-coapproximating systems (Prop 4.6) and by representability (Thm 5.18), giving falsifiable handles within the paper.

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Pith. "Pith review of Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems." pith.science (2026). https://pith.science/paper/FF5WEFUC

@misc{pith2026260214383,
  author       = {Pith},
  title        = {Pith review of: Triangulated categories with a compact silting object, Brown-Comenetz duality and Brown representability theorems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FF5WEFUC}},
  note         = {Machine review of arXiv:2602.14383}
}
abstract

The paper develops a Brown--Comenetz dual framework for Neeman's representability theorems for triangulated categories with a single compact generator (Invent. math., 244:531-616, 2026). Starting from a locally Hom-finite approximable triangulated category, we use the Brown--Comenetz duals of compact objects to construct a triangulated subcategory $\E$, which plays the role of an injective-side analogue of the compact subcategory $\T^c$. We introduce the intrinsic subcategory $\T_c^+$, dual to Neeman's subcategory $\T_c^-$, and characterize its objects by strong $\E$-coapproximating systems and homotopy inverse limits. Under the compact silting hypothesis, we prove Brown representability theorems identifying $(\T_c^+)^{\op}$ with locally finite $\E$-homological functors and $(\T_c^b)^{\op}$ with finite $\E$-homological functors. We also establish localization results for recollements on the Brown--Comenetz side and derive applications to derived categories of finite-dimensional algebras.

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Reference graph

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