{"id":"6a49b2cd-49fb-4cd4-b6a2-de6b550fa44d","arxiv_id":"2507.09286","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable equivalence preserves ω-left approximation dimensions and the Wakamatsu tilting conjecture for Artin algebras without nodes or semisimple direct summands.","lead":"The paper proves that a measurement called the ω-left approximation dimension is preserved when two algebras are stably equivalent, under certain conditions. This lets the authors transfer Wakamatsu tilting modules between the two algebras and shows that an open conjecture about these modules holds on one side exactly when it holds on the other.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5 ignores projective-injective summands: the proof drops P' from M, so the asserted equality fails (or is undefined) when P' is not in addω, even under the paper's stable-equivalence hypotheses.","rationale":"The reader's weakest assumption identifies the no-nodes/no-semisimple condition, which is indeed needed for the cited bijections. But the more immediate flaw is module-level: Theorem 3.5's proof reduces to Y⊕I′ and never accounts for the projective-injective summand P′ of M. Because ν always contains the full direct sum Q of projective-injective Γ-modules, any nonzero Q′ in N is already in addν, forcing the right-hand side to be infinite. The left-hand side need not be infinite unless P′∈addω, and the statement does not impose that. This is not a mere citation or proof gap: the transfer formula as written admits a concrete counterexample under the paper's own stable-equivalence hypotheses. The flaw is repairable by requiring P and Q to be the full direct sums of all indecomposable projective-injective modules, in which case the finite case forces P′=Q′=0 and the omitted P′ is harmless, and by proving the infinite case separately. Since the correction is natural and the intended theorem is likely salvageable, I recommend keeping a CONDITIONAL verdict rather than rejecting outright, but the condition is substantive and should be stated explicitly in the theorem.","tokens_in":13190,"tokens_out":34006,"duration_ms":409555,"concrete_test":"Run an identity-stable-equivalence check in a node-free algebra with two nonisomorphic projective-injective modules, e.g., a product of two copies of a self-injective Nakayama algebra with Loewy length 3. Let F=F′=identity, take ω=P1, M=P2, with ω decomposed as X=I=0, P=P1 and M decomposed as Y=I′=0, P′=P2; choose Q′=P2 so N=P2. Then compute l.appωM and l.appνN with ν=Q=P1⊕P2. If l.appωM is not a positive integer or ∞ (because Hom(P2,P1)=0 forces the first map in any complex to be zero) while l.appνN=∞ (because N∈addν), then Theorem 3.5 as stated fails, and the theorem needs the added hypothesis that P and Q are full direct sums of all indecomposable projective-injective modules, plus a separate treatment of the infinite case.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem as stated is not correct for arbitrary projective-injective direct summands. In Theorem 3.5, M = Y ⊕ I′ ⊕ P′ with P′ projective-injective, but N is defined as F′(Y)⊕F(I′)⊕Q′ and the proof never uses P′; it really proves a statement about Y⊕I′ only. Since ν always contains Q, the direct sum of all projective-injective Γ-modules, any nonzero Q′ lies in addν, and any module with a nonzero addν-summand has infinite ν-left approximation dimension: the complex 0→U→U→0→… with identity first map is exact and all maps are left addν-approximations. Thus the right-hand side is automatically ∞ whenever Q′≠0. The left-hand side is not automatically ∞: if P′∉addω, then P′ cannot embed into any module in addω, because an injective module embedded in addω would split off as a direct summand, forcing P′∈addω. Hence no exact complex 0→P′→ω1→… can start, so l.appωM is not positive or infinite. A concrete witness is obtained by taking Λ=Γ to be a product of two copies of a self-injective Nakayama algebra with Loewy length 3 (these have neither nodes nor semisimple direct summands), F=F′=identity, ω=P1, M=P2 for two nonisomorphic projective-injective modules in different blocks, and Q′=P2. Then ν=Q=P1⊕P2 and N=P2; the left side is not positive while the right side is ∞. The intended repair is to require that P and Q are the full direct sums of all indecomposable projective-injective modules of Λ and Γ, as is done in Theorem 3.12, and to prove the infinite-dimensional case separately; Theorem 3.5 and Theorem A only write 'a projective-injective module'.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the behaviour of ω-left approximation dimensions under stable equivalence of Artin algebras having neither nodes nor semisimple direct summands. Under the standing stable equivalence F and its companion F′, the authors define, for ω=X⊕I⊕P, the module ν=F′(X)⊕F(I)⊕Q, where Q is the direct sum of all indecomposable projective-injective Γ-modules, and claim in Theorem 3.5 that for M=Y⊕I′⊕P′ and N=F′(Y)⊕F(I′)⊕Q′ one has l.app_ω M = l.app_ν N. The paper then derives transfer results for faithful dimension and dominant dimension, gives one-to-one correspondences between basic (Wakamatsu) tilting modules, proves that the Wakamatsu tilting conjecture is preserved under stable equivalence, and formulates analogous statements for relative n-torsionfree modules and generalized Gorenstein dimension.","tokens_in":13545,"tokens_out":15585,"duration_ms":188600,"significance":"If the main theorem were correct, the paper would supply a useful transfer principle for a homological invariant under stable equivalence, with concrete consequences for the Wakamatsu tilting conjecture and for relative homological dimensions. The strategy is well motivated: it uses established stable-equivalence machinery of Auslander–Reiten and Martinez-Villa, and the explicit formulas for ν in terms of ω are natural. The paper is also honest in relying on external theorems rather than inventing ad hoc axioms. However, the central theorem mishandles the projective-injective summand P′, and the infinite-dimensional case of the approximation dimension is not proved; these issues affect the faithful-dimension and tilting applications. The results would be significant after a careful correction of the statement and proof of Theorem 3.5.","major_comments":[{"comment":"The projective-injective summand P′ plays no role in the proof: the proof constructs an exact sequence for N starting from a resolution of Y only, appending F(I′)⊕Q′ as a split summand, and it never uses P′ or relates Q′ to F′(P′). This is a load-bearing omission. If P′ is not in addω, then no left addω-approximation of P′ can be injective, because an injective submodule of a module in addω splits off and would force P′∈addω; hence no exact truncated complex can start at M with terms in addω, and l.app_ω M is not positive/infinite. On the other hand, Q′ is a projective-injective Γ-module, so Q′∈addν because ν contains the full direct sum Q of all such modules; taking M=P′ and N=Q′ with P′∉addω and Q′≠0 gives l.app_ν N=∞ while l.app_ω M is not positive/infinite, contradicting the asserted equality. A concrete witness is obtained by taking Λ=Γ to be a product of two copies of a suitable self-injective Nakayama algebra, F=F′=id, ω=P1, M=P2 for two nonisomorphic projective-injective modules in different blocks, and Q′=P2. The theorem needs either corrected hypotheses (for example, that P and Q are the full direct sums of all indecomposable projective-injective modules and that Q′ is tied to P′) or a separate argument showing how P′ and Q′ are eliminated; the current proof does not establish either.","section":"§3.1, Theorem 3.5"},{"comment":"The proof treats only the cases l.app_ω M = 1 and l.app_ω M = n for a positive integer n; the case l.app_ω M = ∞ is not addressed. This is not a cosmetic omission, because Proposition 3.7 and Proposition 3.9 need the equality of faithful dimensions in the infinite case to conclude that a Wakamatsu tilting module is transferred to a Wakamatsu tilting module. Since Lemma 3.4(1) is formulated for finite n, an additional argument is required to show that an infinite exact complex with left addω-approximations is transferred to an infinite exact complex with left addν-approximations; the present proof only gives arbitrarily long finite exact complexes, whose compatibility is not shown.","section":"§3.1, proof of Theorem 3.5"},{"comment":"Lemma 3.4(2) is stated with the sentence 'the proof of (2) is similar' and is used in Proposition 3.10 to transfer the resolving sequence 0→Λ→ω0→…→ωn→0 to Γ. This is a different statement from part (1): it requires a finite exact sequence ending at 0 with all middle terms in addν, not merely a left-approximation sequence of finite length. The truncation and induction used in part (1) do not automatically give the required terminal exactness. A complete proof or a precise reduction to part (1) should be supplied.","section":"Lemma 3.4(2)"},{"comment":"Theorem 3.11 asserts that Φ and Ψ restrict to one-to-one correspondences between WT(Λ) and WT(Γ), and between T(Λ) and T(Γ), but the proof only cites Propositions 3.9 and 3.10, which show that Φ sends each class into the corresponding class. The verification that these maps are inverse to each other—that is, Ψ(Φ(ω))≅ω and Φ(Ψ(ν))≅ν—is not given. This inverse property is used in Theorem 3.12 to conclude that ω is tilting, so the bijectivity claim needs an explicit proof.","section":"Theorem 3.11"}],"minor_comments":[{"comment":"The statement of Theorem 3.5 defines N=F′(Y)⊕F(I)⊕Q′, but the proof uses F(I′) and requires I′∈addI(Λ)P; the displayed definition should presumably read F(I′).","section":"Theorem 3.5"},{"comment":"The proof of Lemma 3.4(1) contains several local errors and inconsistencies: 'Definie g1' should be 'Define g1', the notation T′1 appears where T1 or T11 is intended, and the displayed formula for the approximation of F′(Y1)⊕F(J1) uses inconsistent subscripts. These should be corrected throughout.","section":"Lemma 3.4(1)"},{"comment":"In the proof of Theorem 3.5, 'left add V -approximation' should be 'left addν-approximation', and the expressions '1.appνN' and 'l .appνN' should consistently be 'l.appνN'.","section":"Theorem 3.5 proof"},{"comment":"In Corollary 3.15, 'Taking ν = F′(Y)⊕F(I′)⊕Q′' should be 'Taking N = F′(Y)⊕F(I′)⊕Q′', and 'projecitve' is a typo for 'projective'.","section":"Corollary 3.15"},{"comment":"References [12], [14], and [20] are all the same arXiv item (Enomoto's 'Maximal self-orthogonal modules and a new generalization of tilting modules') and should be consolidated; [24] is listed as 'preprinted' without a venue or arXiv number.","section":"References"},{"comment":"The phrase 'the Nakamaya conjecture' should be 'the Nakayama conjecture'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a representation theory journal and the intended transfer results would be attractive if the projective-injective summand issue is resolved. The main theorem as stated is false for arbitrary P′, so the revision needs to add correct hypotheses and a revised proof; the infinite-dimensional case and the inverse correspondence in Theorem 3.11 also require attention. The citation list contains duplicate entries and one unpublished preprint; these should be cleaned up before resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the headline theorem is false as stated. The proof of Theorem 3.5 never uses the projective-injective summand P′ of M; it proves the claim only for Y⊕I′. That is not a gap you can patch with a sentence. Take Λ=Γ to be a product of two copies of a self-injective Nakayama algebra of Loewy length 3, F=F′=identity, ω=P1 (one projective-injective), M=P2 (the other), Q′=P2. Then ν=P1⊕P2, N=P2∈addν, so l.app_ν N=∞. But M=P2 is not in addω and is injective, so any exact complex 0→P2→ω1→… would make P2 a direct summand of ω1; impossible. The left side is not positive (or even defined). So the equality l.app_ω M=l.app_ν N fails. The same issue infects Proposition 3.7 and Corollary 3.8 as stated.\n\nWhat is genuinely new is the transfer idea: stable equivalence should preserve ω-left approximation dimensions, faithful dimension, and the Wakamatsu tilting conjecture. The class of algebras (no nodes, no semisimple direct summands) is the right one, and the authors correctly see that the full projective-injective summand of the target algebra has to be added to ν. The applications to Wakamatsu tilting modules are plausible and may survive once the statement is fixed. The intended repair is to require P and Q to be the full direct sums of all projective-injective modules (as in Theorem 3.12) and to handle the infinite-dimensional cases explicitly. Under that repair, Theorem 3.5 might well be true.\n\nOther weaknesses: Lemma 3.4(2) is asserted with 'proof is similar', and Theorem 3.11's inverse correspondence is not checked. The reference list contains duplicates ([12]=[14]=[20]) and some likely mis-citations. These are secondary.\n\nBottom line: this paper should not be accepted as is, but it deserves a serious referee and a major-revision decision. The core idea is right, the execution has a load-bearing mistake that is repairable.","headline":"Theorem 3.5 as stated is false: projective-injective summands are dropped, so the equality fails on an explicit self-injective example; the intended transfer theorem may be repairable but needs major revision.","tokens_in":14101,"tokens_out":10876,"would_cite":false,"duration_ms":128517,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D20","16E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stable equivalence preserves ω-left approximation dimensions and thereby carries the Wakamatsu tilting conjecture between Artin algebras without nodes or semisimple direct summands.","keywords":["ω-left approximation dimension","stable equivalence","faithful dimension","Wakamatsu tilting module","Wakamatsu tilting conjecture","relative n-torsionfree modules","dominant dimension","Artin algebras"],"falsifier":"Find a pair of stably equivalent Artin algebras without nodes and without semisimple direct summands, a self-orthogonal module $\\omega$ with the stated decomposition, and a module $M$ for which direct computation of the left $\\mathrm{add}\\,\\omega$-approximation resolutions gives different $\\omega$-left approximation dimensions on the two sides; the theorem's equality would be false in that case.","tokens_in":12966,"feed_emoji":"🔁","tokens_out":12380,"duration_ms":115469,"temperature":0.7,"pith_summary":"This paper proves that $\\omega$-left approximation dimensions—a measure of how many steps of a module's best approximation complex by direct sums of a fixed module $\\omega$ remain exact—pass unchanged between two Artin algebras that are stably equivalent in the sense of module-category equivalence modulo projectives. The transfer works when the algebras have neither nodes nor semisimple direct summands and the module $\\omega$ is self-orthogonal and decomposed into a module without injective summands, an injective-projective part, and a projective-injective part. If the theorem is right, then stable equivalence is strong enough to preserve faithful dimensions, dominant dimensions, basic (Wakamatsu) tilting modules, and the truth of the Wakamatsu tilting conjecture in this class of algebras. The paper's broader message is that a certain tilting-theoretic invariant is a stable-equivalence invariant.","feed_headline":"Stable equivalence preserves ω-left approximation dimensions","feed_subtitle":"Wakamatsu tilting conjecture and tilting modules transfer between stably equivalent node-free Artin algebras.","key_machinery":"The load-bearing device is the pair of functors $F: \\underline{\\mathrm{mod}}\\,\\Lambda \\to \\underline{\\mathrm{mod}}\\,\\Gamma$ and $F' = \\tau_\\Gamma F \\tau_\\Lambda^{-1}$ that arise from a stable equivalence, together with the module correspondence $\\nu = F'(X) \\oplus F(I) \\oplus Q$ built from the decomposition of $\\omega$. These functors biject the subcategories of modules without projective or injective summands and the subcategories of projective-injective modules (Lemma 3.1), and Lemma 3.3 shows $\\operatorname{Ext}^n_\\Lambda(A,A') \\cong \\operatorname{Ext}^n_\\Gamma(B,B')$ for modules related by this correspondence. Lemma 3.4 is the actual transfer engine: it converts a left $\\mathrm{add}\\,\\omega$-approximation resolution of $M$ into a left $\\mathrm{add}\\,\\nu$-approximation resolution of $N$ term by term, which is exactly what makes the equality of dimensions in Theorem 3.5 go through.","core_discovery":"The central assertion is Theorem 3.5: for stably equivalent Artin algebras $\\Lambda$ and $\\Gamma$ with neither nodes nor semisimple direct summands, given $\\omega = X \\oplus I \\oplus P$ with $\\operatorname{Ext}^1_\\Lambda(\\omega,\\omega)=0$ and a module $M = Y \\oplus I' \\oplus P'$, the module $\\nu = F'(X) \\oplus F(I) \\oplus Q$ on the $\\Gamma$ side satisfies $l.\\mathrm{app}_\\omega M = l.\\mathrm{app}_\\nu N$, where $N = F'(Y) \\oplus F(I') \\oplus Q'$. The proof builds a transfer mechanism in Lemma 3.4 that sends left $\\mathrm{add}\\,\\omega$-approximation sequences to left $\\mathrm{add}\\,\\nu$-approximation sequences, using the fact that the stable equivalence functors $F$ and $F'$ preserve extensions (Lemma 3.3) and biject the relevant subcategories of modules. From this, the paper derives that faithful dimension is preserved (Proposition 3.7), that the maps $\\Phi$ and $\\Psi$ give one-to-one correspondences between basic Wakamatsu tilting modules and between basic tilting modules (Theorem 3.11), and that the Wakamatsu tilting conjecture holds for $\\Lambda$ exactly when it holds for $\\Gamma$ (Theorem 3.12).","pith_inferences":["If the theorem is correct, any invariant expressible through $\\omega$-left approximation dimensions for modules of the stated decomposable form will transfer automatically, so the result may serve as a template for finding new stable-equivalence invariants beyond those listed.","The node-free, semisimple-summand-free hypothesis is exactly where the subcategory bijections used in the proof are available; a natural boundary test is whether a stable equivalence involving an algebra with nodes or a semisimple direct summand can break the equality of approximation dimensions.","Because the correspondence matches Wakamatsu tilting modules one-to-one, the class of algebras satisfying the Wakamatsu tilting conjecture is closed under the relevant stable equivalences; this may suggest that a broader class of equivalences transfer the conjecture as well."],"forward_implications":["Faithful dimension is a stable-equivalence invariant: $\\operatorname{fadim}_\\Lambda \\omega = \\operatorname{fadim}_\\Gamma \\nu$ for the corresponding module $\\nu$ (Proposition 3.7).","Dominant dimension is preserved: $\\operatorname{dom.dim} M = \\operatorname{dom.dim} N$, and in particular $\\operatorname{dom.dim} \\Lambda = \\operatorname{dom.dim} \\Gamma$ (Corollary 3.8).","Basic Wakamatsu tilting modules and basic tilting modules correspond one-to-one between $\\Lambda$ and $\\Gamma$ via $\\Phi$ and $\\Psi$ (Theorem 3.11).","The Wakamatsu tilting conjecture is a stable-equivalence invariant: $\\Lambda$ satisfies it if and only if $\\Gamma$ does (Theorem 3.12).","Relative $n$-torsionfree modules, modules that are $\\omega$-$\\infty$-torsionfree, and modules with generalized Gorenstein dimension zero transfer to the $\\Gamma$ side (Propositions 3.13, Corollary 3.15, Theorem 3.16)."],"supporting_citations":[{"why":"Supplies the stable equivalence functors F and F' and the bijections on module subcategories that carry the transfer.","marker":"[3]"},{"why":"Its Lemma 4.10 gives the one-to-one correspondences for projective-injective modules used in Lemma 3.1.","marker":"[9]"},{"why":"Introduces the omega-left approximation dimension that the paper transfers.","marker":"[17]"},{"why":"Provides the extension-preservation and projective-dimension facts used in Lemma 3.3 and Theorem 3.12.","marker":"[22]"},{"why":"Defines Wakamatsu tilting modules, the objects of the conjecture transferred in Theorem 3.12.","marker":"[25]"},{"why":"Establishes the extension isomorphism for modules of finite projective dimension used in Lemma 3.3.","marker":"[21]"},{"why":"Defines faithful dimension as the omega-left approximation dimension of the regular module, used in Proposition 3.7.","marker":"[11]"},{"why":"Introduces stable equivalence of Artin algebras, the setting of the whole paper.","marker":"[2]"}],"fun_headline_variants":["ω-left approximation dimensions survive stable equivalence","Stable equivalence transfers ω-left approximation dimensions","Tilting dimensions transfer across stable equivalence","Stable equivalence fixes ω-left approximation dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole transfer rests on the pair of algebras being stably equivalent with neither nodes nor semisimple direct summands, and on the module $\\omega$ being self-orthogonal.","fun_headline_variants_meta":{"raw":{"variants":["ω-left approximation dimensions survive stable equivalence","Stable equivalence transfers ω-left approximation dimensions","Tilting dimensions transfer across stable equivalence","Stable equivalence fixes ω-left approximation dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2003,"prompt_tokens":881,"completion_tokens":1122,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1068}},"tokens_in":497,"tokens_out":1122,"duration_ms":13345,"temperature":1.0,"reasoning_tokens":1068,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:00:19.772568+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a pair of stably equivalent Artin algebras without nodes and without semisimple direct summands, a self-orthogonal module $\\omega$ with the stated decomposition, and a module $M$ for which direct computation of the left $\\mathrm{add}\\,\\omega$-approximation resolutions gives different $\\omega$-left approximation dimensions on the two sides; the theorem's equality would be false in that case.","supporting_citations":[{"cited_title":"Auslander, I","cited_arxiv_id":null,"evidence_quote":"Supplies the stable equivalence functors F and F' and the bijections on module subcategories that carry the transfer."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Its Lemma 4.10 gives the one-to-one correspondences for projective-injective modules used in Lemma 3.1."},{"cited_title":"Huang, ω-k-torsionfree modules and ω-n-left approximation dimension, Science in China (Series A)(2000), 44(2): 184-192","cited_arxiv_id":null,"evidence_quote":"Introduces the omega-left approximation dimension that the paper transfers."},{"cited_title":"Martinez-Villa, Properties that are left invariant under stable equivalence","cited_arxiv_id":null,"evidence_quote":"Provides the extension-preservation and projective-dimension facts used in Lemma 3.3 and Theorem 3.12."},{"cited_title":"Wakamatsu, On modules with trivial self-extensions, J","cited_arxiv_id":null,"evidence_quote":"Defines Wakamatsu tilting modules, the objects of the conjecture transferred in Theorem 3.12."},{"cited_title":"Miyashita, Tilting modules of finite projective dimension, Math","cited_arxiv_id":null,"evidence_quote":"Establishes the extension isomorphism for modules of finite projective dimension used in Lemma 3.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines faithful dimension as the omega-left approximation dimension of the regular module, used in Proposition 3.7."},{"cited_title":"Auslander, I","cited_arxiv_id":null,"evidence_quote":"Introduces stable equivalence of Artin algebras, the setting of the whole paper."}],"review_version":1}