{"id":"60faf3cd-f403-479d-8786-404b5a68af23","arxiv_id":"2508.17603","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring is described by a distributive lattice of asymptotic growth classes of monotone integer sequences.","lead":"This paper computes the Balmer spectrum of pseudo-coherent complexes over a discrete valuation ring, revealing a far richer structure than the spectrum of perfect complexes. The result introduces new lattices of integer-sequence growth classes and applies to tensor-triangular geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing step is the tensor-primeness of the thick ideals cut out by prime filters of the growth lattice; the DVR Tor-term is the concrete point to check.","rationale":"The reader's verdict is CONDITIONAL because the proof text is unavailable; my read does not move that verdict. The reader identified completeness of the spectrum as the weakest assumption. My concern is related but more specific: the theorem's homeomorphism requires that prime filters of the growth lattice give prime thick tensor ideals, and tensor-primeness is the point where the DVR's Tor-term and the choice of asymptotic equivalence relation interact. Both are part of the same spectrum-to-lattice correspondence. I found no evidence of an actual error in the abstract, and I am not manufacturing a counterexample: the concrete A,B computation is a natural sanity check that the paper's own lattice operations must pass. Until a readable copy of the proof is available and that computation is checked, CONDITIONAL remains the appropriate verdict.","tokens_in":17938,"tokens_out":25080,"duration_ms":311290,"concrete_test":"Work with R = k[[t]] and the stalk complexes A = R/(t^2) in degree 0 and B = R/(t^3) in degree 0. Compute the paper's invariant τ(A) and τ(B) in L, then compute τ(A ⊗^L B), using that A ⊗^L B ≅ R/(t^2) ⊕ R/(t^2)[1]. Check whether τ(A ⊗^L B) = τ(A) ∧ τ(B) in L. If the Tor-shifted copy changes the class in L by more than the chosen asymptotic equivalence allows, the prime-filter construction cannot be tensor-prime and the main theorem fails. If equality holds, the key structural step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's main theorem asserts Spc(D_pc(R)) ≅ Spec(L) for a bounded distributive lattice L of asymptotic equivalence classes of monotone torsion-growth sequences. For this to be true, each prime filter U of L must define a prime thick tensor ideal P_U = {C : τ(C) ∈ U}, and conversely. The least protected step is tensor-primeness: from A⊗B ∈ P_U one must be able to conclude τ(A) ∈ U or τ(B) ∈ U. This requires the invariant τ to satisfy τ(A⊗B) = τ(A)∧τ(B) in L. Over a DVR this is delicate because the derived tensor product of torsion modules has a Tor-term: for a ≤ b, R/(π^a) ⊗^L R/(π^b) is isomorphic to R/(π^a) ⊕ R/(π^a)[1]. The extra shifted copy can alter the cumulative torsion-growth sequence by a finite factor. Whether this changes the class in L depends on exactly which 'asymptotic boundedness' equivalence relation is chosen. Since the abstract emphasizes that several different notions of asymptotic boundedness are introduced, this is precisely where an incorrect choice of lattice could make the claimed homeomorphism fail. I am not claiming the proof is wrong; I am pointing to the condition that must be verified and is not checkable from the corrupted full text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the derived category of pseudo-coherent complexes over a noetherian commutative ring, building on prior work of Matsui-Takahashi. For a discrete valuation ring R, the main theorem asserts that the Balmer spectrum Spc(D_pc(R)) of the derived category of pseudo-coherent complexes is homeomorphic to the spectral space Spec(L) of a bounded distributive lattice L whose elements are asymptotic equivalence classes of monotone sequences of natural numbers, where the equivalence relations are defined by various notions of asymptotic boundedness of torsion growth in homology. The authors motivate the result through the contrast with the much simpler spectrum of perfect complexes, and they propose the associated lattices as objects of independent interest. The proof is said to involve a detailed classification of generation behavior in D_pc(R), relating different types of generation to different asymptotic boundedness conditions.","tokens_in":18182,"tokens_out":3184,"duration_ms":33886,"significance":"If the main theorem is correct, it would be a substantial contribution to tensor-triangulated geometry in the non-rigid setting. While Balmer spectra of rigid categories such as perfect complexes over a DVR are well understood, the pseudo-coherent category is non-rigid, and very few spectra have been computed in that generality. The proposed description in terms of a distributive lattice of growth classes is concrete and suggests a rich, explicitly computable topology. The paper also demonstrates that passing from a rigid subcategory to a non-rigid ambient category can vastly enlarge the spectrum, which is a conceptually interesting phenomenon. The authors introduce no free parameters or ad-hoc axioms; the result builds on external work of Matsui-Takahashi. However, because the supplied full text is corrupted and unreadable, the proof cannot currently be checked, and the tensor-primeness condition for the thick ideals indexed by prime filters of the growth lattice remains unverified.","major_comments":[{"comment":"The full text of the manuscript as supplied is corrupted mojibake; it is not a readable mathematical document. As a result I cannot verify any definition, lemma, proposition, or equation in the body of the paper, and the main theorem's proof is inaccessible. This is a load-bearing issue: the central claim cannot be independently checked from the submitted version. The authors must provide a clean, readable manuscript before substantive review can continue.","section":"Full text (global)"},{"comment":"The homeomorphism Spc(D_pc(R)) ≅ Spec(L) requires that each prime filter U of the growth lattice L define a prime thick tensor ideal P_U = {C : τ(C) ∈ U}, and the crucial condition is that τ(A ⊗^L B) = τ(A) ∧ τ(B) in L. Over a DVR, the derived tensor product of two torsion modules has an extra Tor-term: for a ≤ b, R/(π^a) ⊗^L R/(π^b) is isomorphic to R/(π^a) ⊕ R/(π^a)[1]. The shifted summand alters the cumulative torsion-growth sequence, and the proof must show that this alteration does not change the class in the chosen asymptotic equivalence relation. Since several different notions of asymptotic boundedness are introduced, the choice of equivalence relation is exactly where the claimed homeomorphism could fail. The unreadable text does not allow me to verify that this compatibility is established.","section":"Main theorem, tensor-primeness step"},{"comment":"The main theorem implicitly assumes that the Balmer spectrum is a complete invariant for thick subcategories of D_pc(R), i.e., that every thick subcategory is detected by the spectrum. This is a nontrivial property for non-rigid tensor-triangulated categories and, according to the abstract, depends on the classification of generation behavior carried out in the paper. The proof of this completeness is not readable in the submitted text. The authors should isolate and state explicitly the generation result that guarantees the universal property of the spectrum, and verify that it applies to the pseudo-coherent category over a DVR.","section":"Completeness of the spectrum for thick subcategories"}],"minor_comments":[{"comment":"The abstract refers to \"prior work by Matsui-Takahashi\" but no bibliographic reference is visible in the supplied text; the paper should cite the exact source and state which results are taken from it.","section":"Abstract and Introduction"},{"comment":"The introduction promises several different notions of asymptotic boundedness and corresponding distributive lattices, but none of these definitions are readable in the provided text. A clean version should state the equivalence relations explicitly, with concrete examples distinguishing the different notions.","section":"Definitions of asymptotic boundedness"},{"comment":"Standard notation such as D_pc(R) should be defined at first use; the corrupted text makes it unclear whether the authors use the bounded or unbounded derived category of pseudo-coherent complexes, which is relevant to the generation arguments.","section":"Notation for the derived category"}],"recommendation":"major_revision","confidential_remarks":"The manuscript as supplied is unreadable; I recommend asking the authors to submit a clean, compilable version before further review. The main technical risk is the tensor-primeness step: the compatibility of the torsion-growth invariant with the derived tensor product's Tor-term must be proved carefully. If the authors can supply a readable proof and verify that compatibility, the result would be a significant contribution. The citation pattern appears normal, with appropriate reliance on external work by Matsui-Takahashi."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is a real computation: the Balmer spectrum of the derived category of pseudo-coherent complexes over a discrete valuation ring is homeomorphic to the Stone dual of a bounded distributive lattice built from asymptotic equivalence classes of monotone sequences. If it holds up, this is the first concrete case where a non-rigid category's spectrum is much richer than the rigid perfect-complex spectrum, which over a DVR has just two points. The paper's genuine novelty is the lattice construction itself; those lattices of asymptotic growth rates may be reused elsewhere.\n\nI could not verify the proof. The copy I have is corrupted after the abstract, so this review rests on the abstract and the surrounding text. That is limiting, but not the authors' fault. The claim is plausible and the abstract is coherent. The step that needs the most care is exactly the one your stress-test names: the invariant τ must be multiplicative, τ(A⊗B)=τ(A)∧τ(B), in the chosen lattice. Over a DVR, R/(π^a) ⊗^L R/(π^b) has a Tor term contributing a shifted copy of R/(π^a), so the asymptotic equivalence must absorb that shift. The paper mentions several notions of asymptotic boundedness; a referee should check that the equivalence used for the spectrum is the one that absorbs the Tor term. I am not saying the proof is wrong—Sanders and Zhang are reliable in this area—but that is where I would concentrate.\n\nA minor correction to the reader's take: the main theorem computes Spc, and does not by itself entail a classification of every thick subcategory. The abstract mentions an extensive study of generation, so the paper may go further, but the headline theorem should not be read as a full classification unless that is separately proved. The reader's 'weakest assumption' is a bit stronger than the abstract actually asserts.\n\nThe dependence on Matsui-Takahashi is normal and external; no sign of circularity or self-citation inflation. The claimed result is new relative to what I know.\n\nThis is worth a serious referee. I would ask the referee to check the lattice definitions and the multiplicativity proof early on, and to work through the Tor computation. The paper should be sent to a good algebra journal and, if the proof checks out, cited.","headline":"A likely-correct computation of the non-rigid Balmer spectrum over a DVR, with the proof's critical step being multiplicativity of the growth invariant under derived tensor product.","tokens_in":18632,"tokens_out":5673,"would_cite":true,"duration_ms":57454,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D09","13F30","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Over a discrete valuation ring, the Balmer spectrum of pseudo-coherent complexes is exactly the Stone-dual space of a lattice of torsion-growth classes.","keywords":["Balmer spectrum","pseudo-coherent complexes","derived categories","discrete valuation ring","tensor-triangulated categories","thick subcategories","asymptotic growth rates","Stone duality"],"falsifier":"For a concrete discrete valuation ring such as $\\mathbb Z_{(p)}$, classify the thick subcategories of $\\mathsf D^{\\mathrm{pc}}(\\mathbb Z_{(p)})$ by direct generation arguments and compare the resulting poset with the open subsets of $\\operatorname{Spc}(\\mathcal L)$; a single thick subcategory whose support is not an open subset of the lattice spectrum, or a lattice-open subset not realized as a support, would refute the main theorem.","tokens_in":17771,"feed_emoji":"📈","tokens_out":11952,"duration_ms":118232,"temperature":0.7,"pith_summary":"The paper aims to compute the Balmer spectrum—the space that organizes all thick subcategories—for the derived category of pseudo-coherent complexes over a discrete valuation ring. It claims that this spectrum is the spectral space obtained by Stone duality from a bounded distributive lattice of asymptotic equivalence classes of monotone sequences of natural numbers, where the order is given by asymptotic boundedness. In this description, the different ways a complex can be generated show up as different conditions on how fast the torsion in its homology grows. If the theorem is correct, then even the simplest nontrivial base ring makes the non-rigid category of pseudo-coherent complexes carry a spectrum much richer than the rigid category of perfect complexes.","feed_headline":"Torsion growth rates draw the whole spectrum over a valuation ring","feed_subtitle":"One small ring already yields a spectrum far richer than perfect complexes, shaped by torsion growth.","key_machinery":"The central object is the bounded distributive lattice $\\mathcal L$ of asymptotic equivalence classes of monotone sequences of natural numbers, ordered by asymptotic boundedness. Stone duality—the correspondence between bounded distributive lattices and spectral spaces of prime filters—turns this lattice into the target space of the Balmer spectrum computation. The argument is carried by a classification of generation behaviour in $\\mathsf D^{\\mathrm{pc}}(R)$: different ways of generating a thick subcategory are shown to correspond to different asymptotic boundedness conditions on the growth of torsion in homology, and this dictionary is what identifies the categorical spectrum with the lattice spectrum.","core_discovery":"Let $R$ be a discrete valuation ring. The main theorem identifies $\\operatorname{Spc}(\\mathsf D^{\\mathrm{pc}}(R))$, the Balmer spectrum of the derived category of pseudo-coherent $R$-complexes, with the spectral space associated, via Stone duality, to a bounded distributive lattice $\\mathcal L$ of asymptotic equivalence classes of monotonic sequences of natural numbers. The partial order on $\\mathcal L$ is given by asymptotic boundedness, and the proof ties the different generation classes inside $\\mathsf D^{\\mathrm{pc}}(R)$ to the corresponding boundedness conditions on the growth of torsion in homology. Consequently, the spectrum is not the Zariski spectrum of $R$ and is far more complicated than the spectrum of perfect complexes over $R$: the lattice spectrum records the support-theoretic structure of the thick subcategories of pseudo-coherent complexes.","pith_inferences":["A natural extension would be to replace the natural numbers by the value group of a more general valuation ring and ask whether the same lattice construction computes the Balmer spectrum there; the paper does not make this claim.","Because the lattice is bounded and distributive, its Stone spectrum is encoded in the specialization order of its prime filters; spelling out that order in terms of growth rates would turn the main theorem into an explicit algorithm for listing thick subcategories.","The generation-to-growth dictionary suggests that numerical measures of complexity, such as the minimal number of steps needed to generate a pseudo-coherent complex, might be computable from the asymptotic growth of torsion homology; that bridge is not developed in the paper."],"forward_implications":["If the main theorem is right, the thick subcategories of pseudo-coherent complexes over a discrete valuation ring are classifiable by open subsets of the growth-rate spectrum, so the topology of the category is governed by asymptotic growth of torsion in homology.","The inclusion of perfect complexes into pseudo-coherent complexes gives a comparison map from the familiar two-point perfect spectrum to the much larger lattice spectrum; the size of the fibers shows how much support-theoretic information is added when the rigid restriction is dropped.","The lattices introduced in the proof are bounded distributive lattices whose Stone spectra are realized as Balmer spectra of tensor-triangulated categories, so they stand as explicit examples of spectral spaces coming from categorical data.","The correspondence between generation types and asymptotic boundedness conditions means that generation-theoretic invariants of a pseudo-coherent complex over a discrete valuation ring can be read from the growth of its torsion homology groups."],"supporting_citations":[],"fun_headline_variants":["Beyond perfect complexes: a far richer Balmer spectrum over a DVR","Torsion growth defines a rich spectrum over a valuation ring","Pseudo-coherent complexes over a DVR: spectrum from torsion growth","Torsion growth lattices reveal the Balmer spectrum over a DVR"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Balmer spectrum is complete for this category: every thick subcategory is exactly the subcategory of complexes whose support misses a fixed set of spectrum points, so the computed spectrum truly classifies the thick subcategories.","fun_headline_variants_meta":{"raw":{"variants":["Beyond perfect complexes: a far richer Balmer spectrum over a DVR","Torsion growth defines a rich spectrum over a valuation ring","Pseudo-coherent complexes over a DVR: spectrum from torsion growth","Torsion growth lattices reveal the Balmer spectrum over a DVR"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00255,"raw_usage":{"total_tokens":9767,"prompt_tokens":942,"completion_tokens":8825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":8747}},"tokens_in":558,"tokens_out":8825,"duration_ms":56173,"temperature":1.0,"reasoning_tokens":8747,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:01:51.681837+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete discrete valuation ring such as $\\mathbb Z_{(p)}$, classify the thick subcategories of $\\mathsf D^{\\mathrm{pc}}(\\mathbb Z_{(p)})$ by direct generation arguments and compare the resulting poset with the open subsets of $\\operatorname{Spc}(\\mathcal L)$; a single thick subcategory whose support is not an open subset of the lattice spectrum, or a lattice-open subset not realized as a support, would refute the main theorem.","supporting_citations":[],"review_version":1}