{"id":"0848274f-e386-4de4-a723-3b1518115290","arxiv_id":"2505.20803","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Homotopically smashing t-structures over noetherian Dynkin path algebras are compactly generated, with a full classification by poset morphisms from Spec(R) to Filt(Nc(Q)).","lead":"The paper proves that every homotopically smashing t-structure in the derived category of a Dynkin quiver path algebra over a commutative noetherian ring is compactly generated, and classifies all compactly generated t-structures in terms of poset maps from the ring's prime spectrum to filtrations of noncrossing partitions. The result extends known theorems for commutative rings and for stable t-structures to the broader t-structure setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Classification and telescope conjecture rest on the unproved lattice-lift compatibility from [CB24]: Proposition 3.5's Hom/Ext orthogonality must survive base change to every residue field, or the projection/gluing bijection collapses.","rationale":"I read the full argument in good faith. The proof of Theorem 4.2 is structured as: classify compactly generated aisles via the lattice-lift projections (Theorem 3.8, Theorem 3.9), show homotopically smashing coaisles are cogenerated by the injective-module lifts (Theorem 3.15), use Theorem 3.14 to identify the induced poset map, and then invoke injectivity of ω to prove that every homotopically smashing t-structure is compactly generated. The local-to-global principle (Theorem 2.2) and the minimality theorem (Theorem 2.3) are proved in the paper, and their arguments are internally consistent under the Koszul degree convention used in Remark 1.15. The comparison with the reader's weakest assumption is direct: the reader identified the lattice-lift machinery of Crawley-Boevey as the main external dependency, and that is also the point I find least secure. The paper cites [CB24, Theorem B] for existence and uniqueness of exceptional lattice lifts and [CB24, Theorem 4.1] for Hom/Ext rank behavior, but it does not spell out the base-change compatibility needed to transfer orthogonality from KQ to every residue field κ(p). Because all later steps reduce to this transfer, a failure there would invalidate the classification and the telescope conjecture as stated. Since [CB24] is a published result and the quoted statements appear to match its content, I do not regard this as an internal contradiction; it is a dependency whose verification is outside the paper. There is one minor internal issue: Definition 1.14 places the Koszul complex in degrees -1 and 0, while Remark 1.15 and the later proofs use the convention with terms in degrees 0 through n and self-duality K(a)[-n]. Taken literally, Definition 1.14 makes the displayed self-duality false, but all substantive uses are consistent with the corrected convention, so this is a typographical error rather than a mathematical flaw. My recommendation is therefore to keep the reader's ACCEPT verdict: the central claim is plausible and well-supported internally, with the acknowledged external dependency on [CB24] being the only load-bearing assumption that a referee should verify explicitly.","tokens_in":26959,"tokens_out":27778,"duration_ms":293391,"concrete_test":"Check the hypotheses of [CB24, Theorem B and Theorem 4.1] on a concrete non-field example: take R=Z, Q=A_3, and the two indecomposables with dimension vectors (1,1,0) and (0,1,1). Construct the unique exceptional free ZQ-lattice lifts via the tree-module matrices from [Rin98], compute Hom_ZQ(eM,eL) and Ext^1_ZQ(eM,eL) with a computer algebra system (Macaulay2 or Sage), and verify they are finitely generated projective Z-modules of the predicted ranks (dim_K Hom_KQ(M,L) and dim_K Ext^1_KQ(M,L) over a field K). Then reduce modulo p=(2) and check that κ(p)⊗_Z eL is the unique exceptional lift of L over F_2 and that the Hom/Ext dimensions over F_2 match the Z-ranks. If either check fails, Theorem 4.2 is unsupported; if both pass, the lattice-lift dependency is validated in a nontrivial case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bijection in Theorem 4.2(1) and the telescope conclusion in (2) both pass through the maps φ and ψ in (3.1), and those maps are inverted using Proposition 3.5: for indecomposable KQ-modules M and L, Hom_RQ(eM,eL)=0 iff Hom_KQ(M,L)=0, and likewise for Ext^1. This proposition is cited to [CB24, Theorem 4.1], which is said to give that both Hom_RQ(eM,eL) and Ext^1_RQ(eM,eL) are finitely generated projective of constant rank equal to the corresponding dimensions over K. The proof then transfers orthogonality of lifts to orthogonality over any residue field κ(p), via Corollary 3.13 and Theorem 3.14, and this is exactly what makes ω injective in Remark 4.1. What is not supplied in the paper is the compatibility statement that for every prime p, the base change κ(p)⊗_R eL is the unique exceptional lattice lift of L over κ(p), nor that the dimensions of Hom_KQ(M,L) and Ext^1_KQ(M,L) are field-independent for the arbitrary fields κ(p) that occur. If for some p the base-changed lattice is not the unique lift, or if the Ext^1 rank changes, then Proposition 3.5 fails and the identification φ(ψ(σ))(p)=σ(p) in the proof of Theorem 3.8 is unsupported. This is a genuine load-bearing assumption because it is imported as a black box rather than proved, and the rest of the argument is built on it. I did not find a competing internal flaw of comparable weight; the sign inconsistency between Definition 1.14 and Remark 1.15 appears to be a typo in the degree convention for the Koszul complex, since all later proofs use the convention of Remark 1.15 consistently.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies t-structures in the derived category D(RQ) of representations of a Dynkin quiver Q over a commutative noetherian ring R. Its central claims are: (1) the compactly generated aisles of D(RQ) are in order-preserving bijection with poset homomorphisms from Spec(R) to the lattice of filtrations of noncrossing partitions of Q; (2) every homotopically smashing t-structure in D(RQ) is compactly generated, resolving the generalized telescope conjecture for these algebras; and (3) over a regular ring R, wide subcategories of mod(RQ) correspond to Spec(R)-indexed families of wide subcategories of mod(KQ). The proof combines a local-to-global principle and a minimality theorem for homotopically smashing coaisles in D(RC), a lattice-lift theory for Dynkin quivers over commutative rings due to Crawley-Boevey, and an appendix giving a field-independent classification of aisles of D(KQ) by filtrations of noncrossing partitions.","tokens_in":27271,"tokens_out":19741,"duration_ms":204138,"significance":"The results, if correct, are significant: they extend Neeman's theorem for commutative noetherian rings and the classifications of [AJS10] and [HN21] to a natural noncommutative family of algebras, and they provide one of the few affirmative answers to the generalized telescope conjecture outside the commutative setting. The paper is largely self-contained and contains detailed proofs of the local-to-global, minimality, and cohomology-determination statements, as well as an appendix that independently records the field-independence of Dynkin aisles. The main weakness is a load-bearing reliance on the lattice-lift construction from [CB24] and, in particular, on a base-change compatibility statement that is used but not stated or proved explicitly.","major_comments":[{"comment":"The proof of Lemma 3.12 applies Proposition 3.5 with R = κ(p), but that proposition does not, as stated, cover the objects κ(p) ⊗_R eM and κ(p) ⊗_R eL. To apply it one must know that these base-changed lattices are the unique exceptional lattice lifts of M and L over κ(p), and that zero-ness of Hom and Ext^1 over κ(p)Q is equivalent to zero-ness over KQ. The concrete construction in Remark 3.4 plausibly supplies the first point, but the proof does not invoke it, and the second point is never stated. This compatibility is load-bearing: it is used in Proposition 3.8 to prove φ(ψ(σ))(p)=σ(p), in Theorem 3.14, and in the injectivity of ω in Remark 4.1, and therefore in Theorem 4.2. Please add an explicit base-change lemma for the lifts of Theorem 3.3, or cite [CB24] for exactly this statement.","section":"Section 3, Proposition 3.5 and Lemma 3.12"},{"comment":"The statement \"Since aisles are independent of the field\" is not a formal consequence of Theorem A.4. Theorem A.4 only gives an abstract order-preserving bijection Aisle(D(FQ)) ≅ Filt(Nc(Q)) for every field F; it does not by itself identify the aisle in D(κ(p)Q) generated by {κ(p) ⊗ gσ(q) | q⊆p} with the aisle in D(KQ) generated by {σ(q)}. This identification is needed to conclude L ∈ σ(p), and it should be proved as part of the base-change compatibility described in the previous comment.","section":"Section 3.1, proof of Proposition 3.8, final step"}],"minor_comments":[{"comment":"The displayed degrees are inconsistent with Definition 1.14: if K(a) lies in degrees [-n,0], its base change to κ(p) should have terms in degrees [-i,0] (or [i-n] as used later in Theorem 3.14), not [i]. The degree convention should be normalized throughout.","section":"Remark 1.15, first bullet"},{"comment":"The symbols M and L silently switch from KQ-modules to their base changes κ(p) ⊗ eM and κ(p) ⊗ eL; this makes the proof of the converse direction unnecessarily hard to parse. Use explicit notation for the base-changed objects.","section":"Lemma 3.12"},{"comment":"Proposition 3.5 is referred to as \"Theorem 3.5\" in the proof of Corollary 3.6 and in Lemma 3.12; please standardize these cross-references.","section":"Corollary 3.6 and Lemma 3.12"},{"comment":"The letter E is used both for the set of modules defining U and for a two-term complex E^0→E^1; distinguishing these would remove avoidable confusion in the proof.","section":"Theorem 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal and the main theorems are attractive. The base-change compatibility of the Crawley-Boevey lattice lifts is the only point that prevents me from recommending acceptance now; it is a gap that should be closed either by a short proof or by a precise citation. I do not think the author needs to reproduce [CB24, Theorem B] itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sabatini proves the generalized telescope conjecture for t-structures in D(RQ) for Dynkin Q and commutative noetherian R: every homotopically smashing t-structure is compactly generated. He also classifies compactly generated aisles as poset maps Spec(R) -> Filt(Nc(Q)). That is a real extension of [AS16] (stable case) and [AJS10]/[HN21] (commutative case), and it unifies them. The paper is clearly written and the main proofs are detailed; I checked the local-to-global principle, the minimality theorem, and the cohomology-determination lemma, and they are consistent.\n\nThe appendix deserves credit too: it gives a self-contained proof that aisles of D(KQ) are classified by filtrations of noncrossing partitions, independent of the field, which I believe was not explicitly recorded.\n\nWhere are the soft spots? The entire classification hinges on Proposition 3.5, which imports from [CB24] that Hom and Ext^1 dimensions of lattice lifts are constant and equal to the field dimensions. The paper does not spell out the compatibility that base-change of the lift to kappa(p) is the unique lift over kappa(p), nor prove field-independence of those dimensions for arbitrary residue fields. That is a genuine dependency, but it is published work, and the tree-module construction makes it very believable. I would not call it a fatal flaw; it is a black box that a referee should ask to be opened or at least stated precisely.\n\nOne minor typo: the Koszul complex degrees in Definition 1.14 and Remark 1.15 disagree; later proofs use the convention in the remark. And Theorem 4.6's proof relies on the stable telescope conjecture (which the paper established earlier) and [ZC17], which is fine.\n\nBottom line: this is a solid contribution that deserves a serious referee. I'd send it to someone comfortable with support theory and t-structures, with a request to scrutinize the [CB24] dependency. I would cite it and probably include it in a reading group.","headline":"A genuinely new positive case of the generalized telescope conjecture for t-structures, well argued and worth refereeing; the main risk is the unproved compatibility with Crawley-Boevey's lattice lifts.","tokens_in":27903,"tokens_out":2827,"would_cite":true,"duration_ms":28374,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E35","16G20","16G30","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the generalized telescope conjecture for derived categories of noetherian Dynkin path algebras and classifies compactly generated t-structures by poset maps into noncrossing-partition filtrations.","keywords":["derived category","telescope conjecture","t-structure","wide subcategory","Dynkin quiver","noncrossing partitions","noetherian path algebra","compactly generated"],"falsifier":"Run the paper's Example 4.7 concretely: enumerate all order-preserving maps $\\mathrm{Spec}(k[[x]]) \\to \\mathbf{Nc}(A_2)$ and all wide subcategories of $\\mathrm{mod}(k[[x]]A_2)$, then check that gluing each map produces a distinct wide subcategory and that every wide subcategory arises this way; any single mismatch would falsify Theorem 4.6 and the gluing method behind Theorem 4.2.","tokens_in":26691,"feed_emoji":"🔭","tokens_out":11625,"duration_ms":114275,"temperature":0.7,"pith_summary":"This paper proves the generalized telescope conjecture for derived categories of path algebras $RQ$ of Dynkin quivers $Q$ over commutative noetherian rings $R$: every homotopically smashing t-structure, meaning one whose coaisle is closed under directed homotopy colimits, is compactly generated. It also gives a complete classification of compactly generated t-structures, identifying them with order-preserving maps from the prime spectrum $\\mathrm{Spec}(R)$ to the lattice $\\mathrm{Filt}(\\mathbf{Nc}(Q))$ of filtrations of noncrossing partitions of the quiver. The classification works by projecting an aisle to each fibre $D(\\kappa(\\mathfrak{p})Q)$ over a prime ideal and gluing the resulting aisles back along $\\mathrm{Spec}(R)$. When $R$ is regular, the paper additionally classifies wide subcategories of finitely generated $RQ$-modules as poset maps from $\\mathrm{Spec}(R)$ into noncrossing partitions. If correct, the results unify the known commutative-noetherian and Dynkin-algebra classifications and settle an open question posed in [BH21] and [HN21] for this class of algebras.","feed_headline":"Telescope conjecture proved for noetherian Dynkin path algebras","feed_subtitle":"Paper classifies compactly generated t-structures there by poset maps into filtrations of noncrossing partitions.","key_machinery":"The load-bearing machinery is the lattice lift supplied by [CB24]: each indecomposable module over a field fibre lifts uniquely to an exceptional $RQ$-lattice, i.e. a module that is finitely generated projective at every vertex, rigid, and with endomorphism ring $R$, and Hom and $\\mathrm{Ext}^1$ orthogonality of indecomposables is detected by these lifts. Around this sit two structural principles proved for homotopically smashing cosuspended subcategories of $D(RC)$: the local-to-global principle, which reconstructs any object from its stalks $\\Gamma_{\\mathfrak{p}}Y$ using extensions, coproducts, and homotopy colimits, and the minimality of stalk subcategories, which identifies $\\Gamma_{\\mathfrak{p}}Y$ as the closure of $\\mathrm{RHom}_R(\\kappa(\\mathfrak{p}), Y)$. The classification itself is carried by the projection and gluing assignments $\\varphi$ and $\\psi$ between $\\mathrm{Aisle}_{\\mathrm{cg}}(D(RQ))$ and $\\mathrm{Hom}_{\\mathrm{Pos}}(\\mathrm{Spec}(R), \\mathrm{Aisle}(D(KQ)))$, with $\\mathrm{Filt}(\\mathbf{Nc}(Q))$ parametrizing the quiver side independently of the field.","core_discovery":"The central discovery is that the lattice of compactly generated aisles of $D(RQ)$ is isomorphic to $\\mathrm{Hom}_{\\mathrm{Pos}}(\\mathrm{Spec}(R), \\mathrm{Filt}(\\mathbf{Nc}(Q)))$, and that this lattice is in fact the whole lattice of homotopically smashing aisles. The proof associates to an aisle $U$ the map $\\mathfrak{p} \\mapsto \\mathrm{aisle}_{KQ}\\langle L \\in \\mathrm{ind}(D(KQ)) \\mid R/\\mathfrak{p} \\otimes^L_R \\widehat{L} \\in U\\rangle$, where $\\widehat{L}$ is the unique exceptional lattice lifting the indecomposable $KQ$-module $L$; conversely, a poset map $\\sigma$ gives the aisle generated by all $R/\\mathfrak{p} \\otimes^L_R \\widehat{\\sigma(\\mathfrak{p})}$. Two structural results for homotopically smashing cosuspended subcategories, the local-to-global principle and minimality of stalk subcategories, make this projection/gluing correspondence bijective and force every homotopically smashing t-structure to be compactly generated. In the regular case, cohomological determination of aisles yields a further bijection between wide subcategories of $\\mathrm{mod}(RQ)$ and $\\mathrm{Hom}_{\\mathrm{Pos}}(\\mathrm{Spec}(R), \\mathbf{Nc}(Q))$.","pith_inferences":["Beyond the paper's claims, the same projection/gluing scheme would classify homotopically smashing t-structures over any finite quiver once minimality of stalk subcategories is available; the paper's Theorem 2.3 is stated for general small categories and a companion result in this direction is cited.","Beyond the paper's claims, cohomological determination of compactly generated aisles suggests a torsion-pair style description of all t-structures of $D(RQ)$: each filtration of wide subcategories at every prime should glue into a global t-structure.","Beyond the paper's claims, for regular $R$ the bijection with $\\mathrm{Hom}_{\\mathrm{Pos}}(\\mathrm{Spec}(R), \\mathbf{Nc}(Q))$ gives a concrete recipe to compute wide subcategories by gluing residue-field data, which can be tested on examples such as $R = \\mathbb{Z}$ or $R = k[x,y]$ with small Dynkin quivers."],"forward_implications":["The generalized telescope conjecture holds for $D(RQ)$: every homotopically smashing t-structure is compactly generated, answering the question from [BH21] and [HN21] for noetherian Dynkin path algebras.","Every compactly generated t-structure of $D(RQ)$ is classified by an order-preserving map $\\mathrm{Spec}(R) \\to \\mathrm{Filt}(\\mathbf{Nc}(Q))$, with the value at each prime recording the t-structure on the residue-field algebra $\\kappa(\\mathfrak{p})Q$ and the global aisle obtained by gluing these fibre aisles.","Compactly generated aisles are determined on cohomology: a complex lies in the aisle exactly when each cohomology module $H^i(X)[-i]$ lies in the aisle.","When $R$ is regular, wide subcategories of $\\mathrm{mod}(RQ)$ are in bijection with poset maps $\\mathrm{Spec}(R) \\to \\mathbf{Nc}(Q)$, recovering the known classifications over commutative rings and over Dynkin algebras as special cases.","The quiver-side lattice $\\mathrm{Aisle}(D(KQ))$ is independent of the field $K$ and equals $\\mathrm{Filt}(\\mathbf{Nc}(Q))$, a fact recorded explicitly in the appendix."],"supporting_citations":[{"why":"Supplies the lattice-lift theorem and Hom/Ext orthogonality detection used as Theorem 3.3 and Proposition 3.5, on which the projection and gluing assignments rest.","marker":"[CB24]"},{"why":"Establishes the stable telescope conjecture and the smashing-subcategory classification for $D(RQ)$ in terms of poset maps into noncrossing partitions, the starting point extended here to general t-structures.","marker":"[AS16]"},{"why":"Classifies compactly generated t-structures over commutative noetherian rings by filtrations of specialization-closed subsets, and its Proposition 2.4 is used to manipulate aisles of the form $\\mathrm{aisle}_R\\langle R/\\mathfrak{a}\\rangle$.","marker":"[AJS10]"},{"why":"Proves the generalized telescope conjecture for commutative noetherian rings and supplies the notion of homotopically smashing t-structures together with the candidate cogenerators $E(R/\\mathfrak{p})$.","marker":"[HN21]"},{"why":"Provides the support theory for $D(RC)$, including the functors $\\Gamma_V$ and $\\Gamma_{\\mathfrak{p}}$, stalk subcategories, and the local-to-global properties used throughout.","marker":"[Ste13]"},{"why":"Identifies wide subcategories of $\\mathrm{mod}(KQ)$ with noncrossing partitions $\\mathbf{Nc}(Q)$, used both in the appendix and in the wide-subcategory classification.","marker":"[IT09]"},{"why":"Supplies the bijection between thick subcategories of $D^b$ and wide subcategories used to pass from cohomologically determined aisles to the classification of wide subcategories.","marker":"[ZC17]"},{"why":"Provides the bijection between aisles of $D(KQ)$ and suspended subcategories of $D^c(KQ)$ used in Corollary A.4.","marker":"[ŠP16]"},{"why":"Defines homotopically smashing t-structures and records that compactly generated t-structures are homotopically smashing, which frames the telescope conjecture being proved.","marker":"[SŠV23]"}],"fun_headline_variants":["Telescope conjecture holds for noetherian Dynkin path algebras","Smashing t-structures compactly generated for Dynkin path algebras","Noncrossing partitions classify compactly generated t-structures","Telescope conjecture: smashing t-structures are compact","Poset maps from Spec(R) classify t-structures over path algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification rests on the external result [CB24] that every indecomposable representation over a residue field lifts to a unique rigid representation over $RQ$ built from finitely generated free $R$-modules, with Hom and $\\mathrm{Ext}^1$ orthogonality preserved; if that lifting theorem failed, the projection/gluing bijection and the telescope conclusion would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Telescope conjecture holds for noetherian Dynkin path algebras","Smashing t-structures compactly generated for Dynkin path algebras","Noncrossing partitions classify compactly generated t-structures","Telescope conjecture: smashing t-structures are compact","Poset maps from Spec(R) classify t-structures over path algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001413,"raw_usage":{"total_tokens":5711,"prompt_tokens":956,"completion_tokens":4755,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":4666}},"tokens_in":572,"tokens_out":4755,"duration_ms":48125,"temperature":1.0,"reasoning_tokens":4666,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:49:06.700521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's Example 4.7 concretely: enumerate all order-preserving maps $\\mathrm{Spec}(k[[x]]) \\to \\mathbf{Nc}(A_2)$ and all wide subcategories of $\\mathrm{mod}(k[[x]]A_2)$, then check that gluing each map produces a distinct wide subcategory and that every wide subcategory arises this way; any single mismatch would falsify Theorem 4.6 and the gluing method behind Theorem 4.2.","supporting_citations":[],"review_version":1}