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Telescope conjecture for t-structures over noetherian path algebras

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.20803 v1 pith:SJY7KXBS submitted 2025-05-27 math.RT math.ACmath.RA

Enrico Sabatini This is my paper · ORCID
classification math.RTmath.ACmath.RA MSC 16E3516G2016G3018G80
keywords derivedcategorytelescopeconjecturet-structurewidesubcategoryDynkinquivernoncrossingpartitionsnoetherianpathalgebracompactlygenerated
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\mathrm{Hom}_{\mathrm{Pos}}(\mathrm{Spec}(R), \mathbf{Nc}(Q))$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

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

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 3, Proposition 3.5 and Lemma 3.12] 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.
  2. [Section 3.1, proof of Proposition 3.8, final step] 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.
minor comments (4)
  1. [Remark 1.15, first bullet] 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.
  2. [Lemma 3.12] 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.
  3. [Corollary 3.6 and Lemma 3.12] 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.
  4. [Theorem 4.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the classification and telescope theorem are derived from independent external results and internal proofs, with only non-load-bearing forward self-citations.

full rationale

This paper's derivation chain is self-contained in the sense relevant to circularity. The central external input is the lattice-lift theorem of Crawley-Boevey ([CB24]), imported at Theorem 3.3 and Proposition 3.5; this is a published independent result, not a self-citation, and the paper does not define its main objects in terms of the target classification. Proposition 3.5 states that Hom and Ext^1 orthogonality of lattice lifts follows from [CB24, Theorem 4.1], and this compatibility is then used in Corollary 3.13 and Theorem 3.14 to transfer orthogonality to residue fields; this is a genuine dependency, but it is a dependency on an external theorem, not a reduction of the conclusion to the premise. The local-to-global principle (Theorem 2.2) and minimality of stalk subcategories (Theorem 2.3) are proved within the paper from standard tools ([AJS10], [Hrb20]), and Theorem 3.15 uses them to express homotopically smashing coaisles in terms of injective cogenerators. Theorem 4.2(1) is proved by verifying the two assignments are inverse: Theorem 3.8 shows phi composed with psi is the identity, and the surjectivity argument compares omega_V with omega_{V_sigma} via Theorem 3.14, then uses injectivity of omega from Remark 4.1. Theorem 4.2(2) repeats the same comparison for an arbitrary homotopically smashing t-structure, so it does not assume the telescope conjecture as an input. No fitted parameter is renamed as a prediction, and no aisles are defined by the bijection being proved. The only self-references are the forward citation [HS25], announced after Theorem 2.3 as future work, and mentions of the author's supervisors in acknowledgements; these are not load-bearing elements of any proof. The reliance on [CB24] is an external correctness risk, not circularity, because the cited result is stated with assumptions that do not include the target theorem and is used as a tool.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on the cited lattice-lift results of Crawley-Boevey and the Ingalls-Thomas classification, which the paper does not reprove but treats as external anchors. No free parameters are fitted to data, and no new entities are postulated. The axioms listed are the main external or domain-specific premises the proof depends on.

assumptions (9)
  • domain assumption R is commutative noetherian and Q is a Dynkin quiver
    Throughout the paper; the classification uses Spec(R) and noncrossing partitions of Dynkin quivers, and the proofs rely on finite representation type.
  • domain assumption Lattice lift theorem ([CB24, Theorem B], cited as Theorem 3.3): for every indecomposable KQ-module L there is a unique exceptional free RQ-lattice eL with the same rank vector, and any rigid RQ-lattice is exceptional
    This is the bridge from D(KQ) to D(RQ). It is cited, not proven in the paper, and the assignments in (3.1) require it for the classification to be well defined.
  • domain assumption Orthogonality preservation of lattice lifts ([CB24, Theorem 4.1], used in Proposition 3.5): Hom_RQ(eM,eL) and Ext^1_RQ(eM,eL) are finitely generated projective with constant rank equal to dimensions of Hom_KQ(M,L) and Ext^1_KQ(M,L)
    Used to prove Corollary 3.6 and Lemma 3.12, which are essential for showing that the projection φ and gluing ψ are inverse and that the coaisle cogenerators align.
  • domain assumption Ingalls-Thomas classification of wide subcategories of mod(KQ) by noncrossing partitions ([IT09, Theorem 1.1])
    Used in Corollary A.4 to identify Wide(KQ) with Nc(Q), and in Theorem 4.6. This is a published external theorem.
  • domain assumption Shift and t-structure classification for Dynkin algebras: Aisle(D(KQ)) is in bijection with Susp(Dc(KQ)) via [ŠP16, Theorem 4.5(i)]
    Used in Corollary A.4 to pass from suspended subcategories of compact objects to aisles. These are published results.
  • domain assumption Pure-semisimplicity of D(KQ) for Dynkin Q ([Bel00, Theorem 12.20])
    Used in Remark 1.9 to identify coaisles with cosusp^Π and to ensure homotopically smashing coaisles are complete. This is a cited external theorem.
  • domain assumption Dc(RQ) = Db(mod(RQ)) for R regular ([CFH24, 20.2.11])
    Used in Theorem 4.6 to apply the Zhang-Cai bijection. Requires R regular, which is stated in the theorem.
  • domain assumption Zhang-Cai bijection between wide and thick subcategories ([ZC17, Theorem 2.5])
    Used in the proof of Theorem 4.6 to transfer the classification of thick subcategories to wide subcategories. This is a published external theorem.
  • standard math Standard derived-category and support-theory facts (Neeman, Stevenson, Krause, etc.)
    Foundational results used throughout (e.g., compact generation of D(RC), existence of K-injective resolutions, support functors Γ_V and L_V). These are standard and not specific to the paper.

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Pith. "Pith review of Telescope conjecture for t-structures over noetherian path algebras." pith.science (2026). https://pith.science/paper/SJY7KXBS

@misc{pith2026250520803,
  author       = {Pith},
  title        = {Pith review of: Telescope conjecture for t-structures over noetherian path algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJY7KXBS}},
  note         = {Machine review of arXiv:2505.20803}
}
abstract

Let $RQ$ be the path algebra of a Dynkin quiver $Q$ over a commutative noetherian ring $R$. We show that any homotopically smashing t-structure in the derived category of $RQ$ is compactly generated. We also give a complete description of the compactly generated t-structures in terms of poset homomorphisms from the prime spectrum of the ring $\mathrm{Spec}(R)$ to the poset of filtrations of noncrossing partitions of the quiver $\mathrm{Filt}(\mathbf{Nc}(Q))$. In the case that $R$ is regular, we also get a complete description of the wide subcategories of the category $\mathrm{mod}(RQ)$.

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