{"id":"f8522cc1-d217-46a4-b379-d256268e137d","arxiv_id":"2602.14383","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For triangulated categories with a compact silting object, the Brown–Comenetz duals of compact objects form a subcategory E, and the new intrinsic subcategory T_c^+ represents exactly the locally finite E-homological functors.","lead":"This paper builds a mirror/dual version of Neeman's representability theorems for triangulated categories, using Brown–Comenetz duality to construct a new subcategory T_c^+ and prove which homological functors are representable. It matters for algebraists because it completes the compact/non-compact duality picture for derived categories and recollements, with concrete applications to finite-dimensional algebras.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.18 rests on an unproved membership claim F∈[E]_4: Proposition 5.7 only produces a representing object F, and the proof then asserts without argument that F lies in [E]_4, which is needed to apply Lemma 5.17.","rationale":"The reader's verdict flagged two weak points: F∈[E]_4 in Theorem 5.18 and t-exactness of i_* in Theorem 6.5(2). I agree with both, but I regard the F∈[E]_4 gap as the single most load-bearing concern because it directly affects the paper's central representability theorems (Theorem 5.18 and 5.22), whereas the t-exactness issue primarily affects the localization theorem in Section 6. The reader's rationale mentions F∈[E]_4, but the stated weakest assumption is the t-exactness; hence my agreement is partial. My own reading confirms that Theorem 5.18 depends on an unsupported categorical-weight assertion: the objects produced by Proposition 5.7 are not shown to lie in the product-closed subcategory [E]_4. This is not a disagreement with the theorem's possible truth, but the proof as written is incomplete. The t-exactness concern is also valid: the line \"Because i_* is t-exact\" in Theorem 6.5(2) introduces an unproved hypothesis that does not follow from the stated assumptions. Both issues warrant the conditional verdict, so I recommend no change to the reader's verdict.","tokens_in":31737,"tokens_out":11964,"duration_ms":119608,"concrete_test":"Independently reproduce the F constructed in Proposition 5.7 for a minimal test case: take T=D(k[x]/(x^2)), G=A, E=K^b(A-inj), and choose a locally finite E-homological functor such as H=Hom_T(X,−)|_E for an explicit X∈T_c^+. Compute the sequence F_i from Lemma 5.5, form F=Holim F_i, and check directly whether F is isomorphic to an object obtained from E by finitely many products and extensions (i.e., whether F∈[E]_4). If this fails, the proof of Theorem 5.18 collapses; if it holds, write out the induction showing that Proposition 5.7 yields the asserted [E]_4 membership.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central representability theorem (Theorem 5.18, and hence Theorem 5.22) has a load-bearing gap at the step where the proof says: \"Moreover, by the proof of Lemma 5.5 together with Proposition 5.7, we have F∈[E]_4.\" This membership is essential: after Lemma 5.16 supplies a triangle D→F→F_4 with F_4∈T_c^+ and g∈J_4, the proof needs Lemma 5.17 with C=F to conclude Hom_T(g,F)=0 and hence that F is a direct summand of F_4. Lemma 5.17 requires F∈[E]_4. No derivation of this membership is given. Lemma 5.5 only constructs objects F_i admitting strong ⟨E⟩_i-coapproximating systems; that is a different and weaker statement than belonging to [E]_i. In the base case F_1 is built as an infinite coproduct of shifted copies of E, whereas [E]_1 is defined by replacing coproducts with products, so the gap is not merely a missing line. The homotopy-limit construction of Proposition 5.7 might indeed produce an object in [E]_4, but the necessary closure properties are never proved. As written, the proof of Theorem 5.18 does not establish that every locally finite E-homological functor is represented by an object of T_c^+.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":32102,"tokens_out":3931,"duration_ms":41126,"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":[{"comment":"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.","section":"Theorem 5.18, proof (Section 5.3, line after Proposition 5.7)"},{"comment":"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.","section":"Theorem 6.5(2), proof (Section 6.1)"},{"comment":"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.","section":"Lemma 5.4 and its uses (Sections 5.1–5.2)"}],"minor_comments":[{"comment":"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).","section":"Section 2, Definition 2.1 (after preferred t-structures)"},{"comment":"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.","section":"Example 5.8(2)"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section 5, Definition 5.15"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a promising framework and the main statements are plausible, but the two load-bearing gaps—the membership F∈[E]_4 in Theorem 5.18 and the t-exactness of i_* in Theorem 6.5(2)—need to be repaired before the results can be trusted. The missing dual proof of Lemma 5.4 also deserves attention. I do not see grounds for outright rejection, but the current version is not yet acceptable as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper deserves a serious referee, but I would not trust Theorem 5.18 as written. The conditional verdict from the quick read is fair, and the stress-test note lands on a real gap.\n\nWhat is new: the authors construct E as the thick subcategory generated by Brown–Comenetz duals of compact objects, define an intrinsic subcategory T_c^+ that is the formal dual of Neeman's T_c^-, prove a homotopy-limit characterization via strong E-coapproximating systems, and then prove 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 the bounded version is fully faithful with image the finite E-homological functors. If correct, this supplies the missing non-compact half of Neeman's program. The recollement localization results in Section 6 are a natural extension, and the examples for derived categories of finite-dimensional algebras are concrete. The architecture is clear and the amount of detail is substantial.\n\nThe soft spots are in proportion. The main one: in the proof of Theorem 5.18, after Proposition 5.7 produces a representing object F, the proof states \"by the proof of Lemma 5.5 together with Proposition 5.7, we have F∈[E]_4\". That is load-bearing: Lemma 5.17 needs F∈[E]_4 to kill the cophantom map and conclude F is a summand of F_4, hence in T_c^+. Lemma 5.5 only gives objects F_i with strong coapproximating systems; it does not place them in [E]_i. And the base F_1 is built as an infinite coproduct of shifted copies of E, whereas [E]_1 is defined using products. So the gap is not a missing line. Proposition 5.7's homotopy-limit construction might well land in [E]_4 if the right closure properties hold, but those properties are not proved. Until they are, the essential image claim in Theorem 5.18 is unsupported.\n\nSecond, Theorem 6.5(2) relies on i_* being t-exact for the preferred t-structures; the text asserts this without proof or citation. The localization sequence for the Verdier quotients depends on it. This may be true under the compact silting hypotheses, but it needs to be shown.\n\nMinor issues: the reliance on Neeman's [15] as a black box for several lemmas makes verification slow, though that is not itself a flaw. Example 5.8 lists the stable homotopy category as a locally Hom-finite k-linear triangulated category with compact silting object; that claim is at best misleading, since spectra are not naturally k-linear over a field. It is a side remark, not load-bearing.\n\nWho this is for: people working on triangulated categories, representability, and recollements, especially those following Neeman's recent approximability program. I would bring it to a reading group to work through the gap, but I would not cite it in my own work until the authors supply the missing argument. It should go to peer review, not desk rejection.","headline":"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.","tokens_in":32570,"tokens_out":3314,"would_cite":false,"duration_ms":33525,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","16E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["triangulated categories","compact silting object","Brown–Comenetz duality","E-homological functors","representability","recollement","localization","derived categories of algebras"],"falsifier":"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).","tokens_in":31625,"feed_emoji":"🔁","tokens_out":6903,"duration_ms":68965,"temperature":0.7,"pith_summary":"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.","feed_headline":"Brown–Comenetz duals classify finite homological functors","feed_subtitle":"In a category with a compact silting object, the non-compact side is exactly the finite and locally finite E-homological functors.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Brown–Comenetz duals pin finite homological functors","Dual category E classifies finite homological functors","Injective-side analogue yields Brown–Comenetz representability","Compact silting: Brown–Comenetz duals for representability"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Brown–Comenetz duals pin finite homological functors","Dual category E classifies finite homological functors","Injective-side analogue yields Brown–Comenetz representability","Compact silting: Brown–Comenetz duals for representability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1260,"prompt_tokens":766,"completion_tokens":494,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":421}},"tokens_in":510,"tokens_out":494,"duration_ms":4771,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T23:12:34.863178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[],"review_version":1}