{"id":"2240c63b-2558-4cd0-bac5-c26dfefeaf2e","arxiv_id":"2502.09009","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Artin algebras connected by recollements or singular equivalences of Morita type with level l satisfy explicit inequalities on their extension dimensions, with Omega-extension dimension invariant in the singular case.","lead":"This paper proves new upper and lower bounds for the extension dimension, a measure of how far an Artin algebra is from being representation-finite, when two algebras are linked by a recollement of derived module categories or by a singular equivalence of Morita type with level.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5(1) proof cites Lemma 3.3 with the wrong width: it yields w(i^*(A)) where the statement requires w(i_*(B)).","rationale":"The reader's verdict was CONDITIONAL with the main technical concern being the downward-extension hypothesis in Theorem 3.5. My stress-test identifies a more specific proof defect in the same theorem: the cited Lemma 3.3 produces a width attached to the left adjoint of the embedding (w(i^*(A))), whereas the theorem statement uses the width of the embedded generator (w(i_*(B))). These widths need not coincide, so the proof of Theorem 3.5(1)(i), and by extension any part of Theorem 3.5 that relies on the same invocation, is not justified as written. However, this does not undermine the paper's main singular-equivalence theorem (Theorem 4.3), whose proof I checked line by line: the stable isomorphisms, the split exact sequence in B^op-mod, the use of Lemma 2.4(3), and the syzygy-shift argument for Ω-ext.dim all appear internally consistent, modulo the notational issue where 'A' should be 'B' in the displayed class [N⊗_A U ⊕ A]. Since the reader already assigned CONDITIONAL based on technical gaps in Section 3, and my concern is a refinement of that, I do not recommend changing the verdict. The concrete test will determine whether the Theorem 3.5 proof defect is genuine or whether an unstated identification of the widths rescues it.","tokens_in":35798,"tokens_out":56247,"duration_ms":530582,"concrete_test":"For the triangular matrix algebra A = (B M 0 C) with B = k[x]/(x^2), C = k, and M = k (the simple B-module), compute the two widths in the induced recollement D(B) → D(A) → D(C). Verify: (1) w(i_*(B)) = 0 because i_*(B) = A e_1 is a direct summand of A as a left A-module; (2) w(i^*(A)) = pd_B(e_1A) = pd_B(B ⊕ M) = ∞. If these computations are correct, then Lemma 3.3(2) cannot produce the bound in Theorem 3.5(1)(i), since it would give a bound involving the infinite width w(i^*(A)). This settles whether the proof as written is valid; if the widths are instead equal, or if a different adjunction is intended, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.5(1), the authors claim that (i) ext.dim(B) ≤ ext.dim(A) + w(i_*(B)) follows from Lemma 3.3(2). But Lemma 3.3(2) applies to a fully faithful functor G : D(B) → D(A) and its left adjoint F : D(A) → D(B), giving ext.dim(B) ≤ ext.dim(A) + w(F(A)). For the recollement inclusion G = i_* : D(B) → D(A), the left adjoint is F = i^* : D(A) → D(B), so the width produced by the lemma is w(i^*(A)), not w(i_*(B)). These two widths are not obviously equal, and for a triangular matrix algebra A = (B M 0 C) one expects w(i_*(B)) = 0 (since A e_1 is projective) while w(i^*(A)) = pd_B(e_1A) can be positive or infinite. Thus the cited application does not establish the stated inequality as written. The same type of adjunction/width mismatch may affect other parts of Theorem 3.5 where Lemma 3.3 is invoked with 'analogous argument'. This is a proof-level gap in the recollement results; it does not appear to affect Theorem 4.3, whose proof is independent and seems sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the behavior of extension dimensions of Artin algebras under two types of links between derived module categories: recollements and singular equivalences of Morita type with level. The authors introduce homological width for triangle functors and use it to state inequalities comparing extension dimensions of algebras in a recollement that extends one step downwards (and upwards), including bounds involving the widths of the induced functors. They also prove that a singular equivalence of Morita type with level l implies |ext.dim(A) − ext.dim(B)| ≤ l, along with invariance of the Ω-extension dimension and of the extension dimension of the category of infinite syzygies. Several corollaries are derived for triangular matrix algebras, homological epimorphisms, exact contexts, and bounded extensions of algebras.","tokens_in":36059,"tokens_out":11836,"duration_ms":107215,"significance":"The results are potentially valuable: extension dimension measures how far an algebra is from being representation-finite, and an invariance statement under singular equivalences with level, as in Theorem 4.3, would be a clean and useful addition. The proof of Theorem 4.3 is detailed and largely self-contained, using standard syzygy and stable-category techniques. The recollement inequalities in Theorem 3.5, if correct, would provide a flexible tool for bounding extension dimensions. However, the central recollement proof contains a width-mismatch in an application of Lemma 3.3; this is a load-bearing gap that must be repaired before the recollement claims are supported. The paper also makes extensive use of 'analogous arguments,' which complicates verification of the remaining parts of Theorem 3.5. The Section 4 contribution appears sound and is the strongest part of the manuscript.","major_comments":[{"comment":"The proof of (1)(i) says that it follows directly from Lemma 3.3(2). Lemma 3.3(2) requires a fully faithful functor G : D(B) → D(A) with left adjoint F : D(A) → D(B), and yields ext.dim(B) ≤ ext.dim(A) + w(F(A)). In the recollement (3.29), the fully faithful functor from D(B) into D(A) is i_*, whose left adjoint is i^*. Thus the lemma produces the term w(i^*(A)), not the stated w(i_*(B)). These two widths are not generally equal: for the canonical recollement of a triangular matrix algebra A = (B M \\ 0 C), i_*(B) is the projective module Ae1, so w(i_*(B)) = 0, while w(i^*(A)) = pd_B(e1A) can be positive or infinite. The same misapplication appears to affect part (1)(ii) and the 'analogous arguments' in Theorem 3.5(2)(i)–(v). The authors should either correct the statement to use the width that the lemma actually yields, or provide a different proof that genuinely gives the stated bound w(i_*(B)).","section":"§3, Theorem 3.5(1)(i)"},{"comment":"The proof claims that Ω^p_D(F(P^•)) = 0 for p ≥ −inf(F) − inf(G). However, F(P^•) is shown to lie in K^{[−sup(G)+inf(G)+1, −inf(G)+sup(G)]}(A-proj), whose width is controlled by w(G), not by w(F). The claimed vanishing bound therefore appears stronger than what the displayed interval justifies; the required bound should plausibly involve w(G). This step is load-bearing because the isomorphism (3.12) is used to transfer the syzygy filtration from F(X) to Y, and the same pattern recurs in later arguments. Please clarify or correct this estimate.","section":"§3, Lemma 3.3(2), near Eq. (3.11)-(3.12)"}],"minor_comments":[{"comment":"There are several typographical errors: 'nonegative' should be 'nonnegative', 'provied' should be 'provided', 'trianle' should be 'triangle', and in the definition of A-inj it says 'all finitely generated projective A-modules' where 'injective' is clearly intended.","section":"§2, Preliminaries"},{"comment":"The proof contains an apparent misprint: 'inf(F(D(A)) = −t' is missing a closing parenthesis, and the final displayed inequality reads 'cw(G(D(B))) ≥ cw(G(D(B)))', which is tautological; presumably one occurrence should be cw(F(D(A))).","section":"§2, Lemma 3.1(2)"},{"comment":"The displayed bounds contain a typographical mismatch in the max expression: 'max{w(i∗(B), w(j∗(A))}' has an extra closing brace and missing a parenthesis; the intended expression is max{w(i_*(B)), w(j_*(A))}.","section":"§1 and §3, Theorem statements"},{"comment":"Several steps are delegated to 'an analogous argument' (e.g., (3.46), (3.49), (3.50)), and at least one of those analogues inherits the same adjunction/width issue flagged in the major comments. The authors should spell out these arguments or clearly state which adjoint is being used in each case.","section":"§3, Theorem 3.5"}],"recommendation":"major_revision","confidential_remarks":"The Section 4 results, especially Theorem 4.3, appear to be the strongest contribution and their proofs seem sound. The recollement section, however, needs genuine repair: the cited application of Lemma 3.3 in Theorem 3.5(1)(i) does not produce the stated width. If the intended statements actually involve w(i^*(A)) and w(j^*(A)), the theorems should be restated accordingly and the applications adjusted. Given the number of 'analogous arguments', a full rewrite of the affected proofs would greatly help verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The singular-equivalence result, Theorem 4.3, is the real payoff: a singular equivalence of Morita type with level l gives |ext.dim(A)-ext.dim(B)| ≤ l and invariance of Ω-ext.dim and ext.dim(Ω^∞). The proof is mostly self-contained, and the use of Heller's theorem to control projective summands is sound. The recollement half, Theorem 3.5, is more fragile. The inequalities are plausible and would be useful if proven, but the proof as written has an adjunction/width mismatch. In (1)(i), Lemma 3.3(2) is applied with G = i_*, so the lemma returns the width w(i^*(A)), not the stated w(i_*(B)). For a triangular matrix algebra these widths are not equal: i_*(B) = Ae1 is projective, width 0, while i^*(A) = e1A can have infinite projective dimension as a B-module. So the cited lemma does not prove the stated inequality. The same kind of mismatch may affect the 'analogous argument' steps in (1)(ii) and (2). Also, Lemma 3.3 has a small overstrong vanishing claim; the later argument seems to use a stronger bound, so that one is likely harmless.\n\nWhat is genuinely new: the level-l bound for singular equivalences extends the authors' earlier l = 0 case, and the recollement inequalities with homological widths are new statements. The paper is careful about not claiming more than it proves, except for the gap above. Citation pattern is fine: self-citations are for auxiliary lemmas and prior work, not for the main theorems.\n\nWho it is for: people working on extension dimensions, Igusa-Todorov distances, and recollement techniques for Artin algebras. Theorem 4.3 alone is worth a cite; the recollement part needs a check.\n\nRecommendation: send it to peer review, but make the referee look hard at the adjunction issue in Theorem 3.5 and at the 'analogous' steps. If the recollement bounds cannot be repaired, the paper still has a solid Section 4. My own verdict: conditional, leaning positive.","headline":"Theorem 4.3 is a solid new result; Theorem 3.5 has an adjunction/width mismatch that needs fixing before the recollement bounds are accepted.","tokens_in":36596,"tokens_out":6850,"would_cite":true,"duration_ms":60645,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E05","16G10","16E10","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two finite-dimensional algebras related by a singular equivalence of Morita type with level $l$ have extension dimensions that differ by at most $l$, and identical asymptotic syzygy dimensions.","keywords":["extension dimension","singular equivalence of Morita type with level","recollement","Artin algebra","derived category","syzygy","homological width","Omega-extension dimension"],"falsifier":"Take a finite-dimensional algebra $A$ and a homological ideal $I$ whose bimodule projective dimension $d := \\mathrm{pd}(A^e A/I)$ is finite, compute $\\mathrm{ext.dim}(A)$ and $\\mathrm{ext.dim}(A/I)$, and check whether $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(A/I)| > 2d$, which would refute Theorem 4.3. Similarly, for any recollement of derived module categories that extends one step downwards, compare $\\mathrm{ext.dim}(A)$ with $2\\,\\mathrm{ext.dim}(B) + \\mathrm{ext.dim}(C) + \\max\\{w(i^*(B)), w(j^*(A))\\} + 2$; a single recollement violating this inequality would refute Theorem 3.5(1)(iii).","tokens_in":35581,"feed_emoji":"🔗","tokens_out":13130,"duration_ms":104904,"temperature":0.7,"pith_summary":"The paper proves quantitative control on extension dimensions, the invariant that measures how far an Artin algebra is from being representation-finite and vanishes exactly for representation-finite algebras. Its central result is that if two finite-dimensional algebras are singularly equivalent of Morita type with level $l$, then their extension dimensions differ by at most $l$, while the asymptotic invariants $\\Omega$-ext.dim and $\\mathrm{ext.dim}(\\Omega^\\infty(\\mathrm{mod}))$ are exactly equal. For recollements of derived module categories, it establishes a parallel family of inequalities: the extension dimensions of the outer algebras are bounded by that of the middle algebra plus a homological width term, and the middle algebra's extension dimension is bounded by a combination of the outer ones, provided the recollement extends one step downward (and, for the sharper bounds, one step upward as well). These bounds matter because they turn loose structural relationships between algebras into concrete constraints on a hard-to-compute invariant.","feed_headline":"Dimension gap ≤ level for singularly equivalent algebras","feed_subtitle":"Two algebras linked at level l have extension dimensions within l of each other, and identical asymptotic ones.","key_machinery":"The argument is carried by three objects. The extension dimension $\\mathrm{ext.dim}(A) = \\inf\\{m \\ge 0 \\mid A\\text{-mod} \\subseteq [M]_{m+1}\\}$ for some module $M$, where $[M]_n$ is the iterated extension closure, measures how many steps of extensions are needed to build all modules from one. The homological width $w(X^\\bullet)$ of a bounded complex of projectives is the minimum length of the interval over which its nonzero terms sit, and for modules of finite projective dimension it equals the projective dimension. The singular equivalence of Morita type with level $l$ is a pair of bimodules $(M,N)$ with $M\\otimes_B N \\simeq \\Omega^l(A)$ and $N\\otimes_A M \\simeq \\Omega^l(B)$ in the stable categories of bimodules. The proof of Theorem 4.3 funnels every $B$-module $Y$ through the filtration of $M \\otimes_B Y$, applies $N\\otimes_A -$ to translate the level-$l$ bimodule isomorphism into $\\Omega^l_B(Y) \\oplus$ projective, and then uses an inverse-syzygy lemma (Lemma 2.4(3)) to pull membership in an extension class back from $\\Omega^l_B(Y)$ to $Y$ itself, at the cost of adding $l$ to the extension dimension.","core_discovery":"The paper's main finding is Theorem 4.3: if $A$ and $B$ are finite-dimensional algebras and $(M,N)$ is a singular equivalence of Morita type with level $l$, meaning $M\\otimes_B N \\simeq \\Omega^l(A)$ and $N\\otimes_A M \\simeq \\Omega^l(B)$ in the stable categories of bimodules, then $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(B)| \\le l$, $\\Omega$-ext.dim$(A) = \\Omega$-ext.dim$(B)$, and $\\mathrm{ext.dim}(\\Omega^\\infty(A\\text{-mod})) = \\mathrm{ext.dim}(\\Omega^\\infty(B\\text{-mod}))$. For recollements of derived module categories (Theorem 3.5), it establishes inequalities such as $\\mathrm{ext.dim}(B) \\le \\mathrm{ext.dim}(A) + w(i^*(B))$ and $\\mathrm{ext.dim}(A) \\le \\mathrm{ext.dim}(B) + \\mathrm{ext.dim}(C) + \\max\\{w(i^*(B)), w(j^*(A))\\} + 1$, where $w(-)$ is the homological width of a complex; these require the recollement to extend one step downwards, with the finer ones requiring an upward extension as well.","pith_inferences":["Read as a metric statement, the theorem makes extension dimension a $1$-Lipschitz invariant on the graph whose vertices are algebras and whose edges are level-$l$ singular equivalences; the paper does not pursue this reading, but it suggests defining the distance between algebras as the minimal level of such an equivalence and asking whether the bound is ever attained.","The mechanism behind the bound, pushing a filtration through a tensor product and pulling it back through a syzygy, suggests that the inequality should be tight in families where the level grows, for instance along bounded extensions with growing $p$; computing those extension dimensions would test whether the $l$ in Theorem 4.3 is genuinely necessary.","An unproved converse is implicit: if two algebras have extension dimensions differing by at most one, one might ask whether they are necessarily linked by a level-one singular equivalence or by a recollement of width one; the paper establishes only the forward direction."],"forward_implications":["A homological ideal $I \\subseteq A$ with finite bimodule projective dimension yields $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(A/I)| \\le 2\\,\\mathrm{pd}(A^e A/I)$ and equal $\\Omega$-extension dimensions (Corollary 4.4).","A bounded extension $B \\subseteq A$ with $(A/B)^{\\otimes p} = 0$ yields $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(B)| \\le 2\\,\\mathrm{pd}(B^e A/B) + p - 1$ and equal $\\Omega$-extension dimensions (Corollary 4.5).","In a recollement extending one step downwards, both outer extension dimensions are bounded by the middle one plus the width of the corresponding complex, and the middle one is bounded by $2\\,\\mathrm{ext.dim}(B) + \\mathrm{ext.dim}(C) + \\max\\{w(i^*(B)), w(j^*(A))\\} + 2$ (Theorem 3.5(1)).","When the recollement also extends upwards, the bound improves to $\\mathrm{ext.dim}(A) \\le \\mathrm{ext.dim}(B) + \\mathrm{ext.dim}(C) + \\max\\{w(i^*(B)), w(j^*(A))\\} + 1$, and when one outer algebra has finite global dimension the $\\Omega$-extension dimension of the middle algebra equals that of the other outer algebra (Theorem 3.5(2)).","Derived equivalences are a special case: for an equivalence $F : D(A) \\to D(B)$, one gets $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(B)| \\le w(F(A))$ with equal $\\Omega$-extension dimensions (Corollary 3.4)."],"supporting_citations":[{"why":"Supplies the definition of singular equivalence of Morita type with level, the equivalence relation whose level bounds the dimension gap in Theorem 4.3.","marker":"[38]"},{"why":"Introduces extension dimension and the fact that it vanishes exactly for representation-finite algebras, the invariant the paper studies.","marker":"[5]"},{"why":"Heller's theorem on the loop-space functor, used in the proof of Theorem 4.3 to pass from bimodule-level syzygy isomorphisms to syzygies of modules.","marker":"[21]"},{"why":"Introduces homological width of complexes, the quantitative measure appearing in every recollement inequality of Theorem 3.5.","marker":"[10]"},{"why":"Supplies the syzygy filtration lemma (Lemma 2.3) used to decompose modules along exact sequences in extension-dimension estimates.","marker":"[44]"},{"why":"Supplies the associativity bound for extension closures (Lemma 2.1) used throughout the filtration and lifting arguments.","marker":"[45]"},{"why":"Provides the syzygy and restriction lemmas for recollements of derived categories (Lemma 2.17) underpinning the downward-extension analysis.","marker":"[40]"},{"why":"Supplies the stable functor machinery for nonnegative functors (Lemma 2.21), used to transfer syzygy membership across the recollement functors.","marker":"[23]"}],"fun_headline_variants":["Singular equivalence bounds extension dimension gap by level","Extension dim gap ≤ level for singularly equivalent algebras","Level-l singular equivalence: extension dimensions differ by ≤ l","Singular equivalence: ext.dim gap bounded by level"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recollement half of the paper rests on the condition that the recollement extend one step downwards, equivalently that $i^*(B)$ be isomorphic in the derived category to a bounded complex of projective $A$-modules, a hypothesis that fails for many recollements; the singular-equivalence half rests on the existence of the two bimodules $M$ and $N$ satisfying the level-$l$ syzygy identities.","fun_headline_variants_meta":{"raw":{"variants":["Singular equivalence bounds extension dimension gap by level","Extension dim gap ≤ level for singularly equivalent algebras","Level-l singular equivalence: extension dimensions differ by ≤ l","Singular equivalence: ext.dim gap bounded by level"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001148,"raw_usage":{"total_tokens":4717,"prompt_tokens":861,"completion_tokens":3856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":3792}},"tokens_in":477,"tokens_out":3856,"duration_ms":27263,"temperature":1.0,"reasoning_tokens":3792,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:57:11.379667+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a finite-dimensional algebra $A$ and a homological ideal $I$ whose bimodule projective dimension $d := \\mathrm{pd}(A^e A/I)$ is finite, compute $\\mathrm{ext.dim}(A)$ and $\\mathrm{ext.dim}(A/I)$, and check whether $|\\mathrm{ext.dim}(A) - \\mathrm{ext.dim}(A/I)| > 2d$, which would refute Theorem 4.3. Similarly, for any recollement of derived module categories that extends one step downwards, compare $\\mathrm{ext.dim}(A)$ with $2\\,\\mathrm{ext.dim}(B) + \\mathrm{ext.dim}(C) + \\max\\{w(i^*(B)), w(j^*(A))\\} + 2$; a single recollement violating this inequality would refute Theorem 3.5(1)(iii).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the definition of singular equivalence of Morita type with level, the equivalence relation whose level bounds the dimension gap in Theorem 4.3."},{"cited_title":"the representation dimension of artin alge- bras","cited_arxiv_id":null,"evidence_quote":"Introduces extension dimension and the fact that it vanishes exactly for representation-finite algebras, the invariant the paper studies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces homological width of complexes, the quantitative measure appearing in every recollement inequality of Theorem 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the syzygy and restriction lemmas for recollements of derived categories (Lemma 2.17) underpinning the downward-extension analysis."}],"review_version":1}