{"id":"b0956264-bed8-4a0e-816a-ec5c3064c641","arxiv_id":"1908.11130","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For split bounded extension algebras A = B ⊕ M, the algebra A satisfies Han's conjecture if and only if the subalgebra B does.","lead":"A math paper proves a closure property for Han's conjecture, an open problem connecting the size of an algebra's homology to its geometric regularity. It shows that if a finite dimensional algebra is built from a smaller one by attaching a 'bounded' bimodule, the larger algebra satisfies the conjecture exactly when the smaller one does.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. The central claim—closure of Han's conjecture under split bounded extensions—is proved through a coherent chain: a reduced relative bar resolution, a nearly exact sequence, a Jacobi-Zariski long exact sequence, and two equivalence results for Hochschild homology and smoothness. I examined the most technical steps, especially Proposition 3.3 and Proposition 4.2(2), and found the arguments internally consistent. The boundedness conditions are used precisely where the proof needs finiteness of the resolution and vanishing of Tor groups; no step silently requires extra hypotheses. The minor typographical issues in Proposition 3.7 (argument order of Tor and the stray 'v') are harmless because the surrounding text and the reference to Proposition 3.3 remove any ambiguity. There is no circularity with Han's conjecture, and no appeal to unverified external results beyond standard homological algebra. The theorem's scope is exactly as stated, and I see no counterexample or hidden assumption that would shift the verdict.","tokens_in":11673,"tokens_out":53424,"duration_ms":478972,"concrete_test":"Verify the main isomorphism on a concrete split bounded extension: let B = k × k, M = k_{1,2} (the one-dimensional simple bimodule with left action through the first idempotent and right action through the second), so A is the path algebra of the quiver 1 → 2 (upper triangular 2×2 matrix algebra). Compute H_*(A,A) and H_*(B,B) directly from the bar resolution and check that they are isomorphic in all degrees * ≥ 1, as predicted by Proposition 4.2(1) (here n = 2, u = 0, so the threshold is 1). This also validates the spectral sequence computation in Proposition 3.3 for a case where M^{⊗_B 2} = 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After a careful review of the argument in Sections 2–4, I find no load-bearing flaw in the proof of Theorem 4.1. The Jacobi-Zariski long exact sequence (Theorem 3.8) is obtained from the m-nearly exact sequence via a spectral sequence argument, and the boundedness hypotheses on M (Definitions 3.4 and 3.6) are exactly what makes the reduced relative bar resolution finite and the spectral sequence page-one terms vanish in high total degree. The two directions of the smoothness equivalence in Proposition 4.2(2) are both justified: the reduction of projective resolutions from A to B uses one-sided projectivity of M, and the converse uses the finite resolution of any A-module by induced modules, which is made exact by the A-linear contracting homotopy. The only issues I found are typographical: in Proposition 3.7 the Tor term is written with the arguments swapped relative to Proposition 3.3, and 'p ≥ v' should read 'p ≥ n'; neither affects the mathematical conclusion because the intended vanishing follows from the projective dimension bound on M^{⊗_B p}. The theorem does not overclaim: the boundedness assumptions are indeed necessary for the quoted Jacobi-Zariski sequence and the finite resolution argument, so the restriction to split bounded extensions is legitimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies split bounded extensions A = B ⊕ M of finite dimensional algebras over an algebraically closed field, where M is B-tensor nilpotent, has finite projective dimension as a Be-module, and is projective as a left or right B-module. The authors construct a reduced relative bar resolution for such extensions (Theorem 2.3), introduce a notion of nearly exact sequence of chain complexes (Definition 3.1), and use it to obtain a Jacobi-Zariski type long exact sequence in high degrees (Theorem 3.8). The main result (Theorem 4.1) states that, for a split bounded extension A = B ⊕ M of finite dimensional algebras, A satisfies Han's conjecture if and only if B does. The proof compares Hochschild homology in high degrees via the Jacobi-Zariski sequence and compares finiteness of global dimension via induced-module filtrations, so the two directions of the Han property are transferred.","tokens_in":11861,"tokens_out":46660,"duration_ms":458306,"significance":"If correct, the main theorem provides a useful closure property for Han's conjecture: it shows the conjecture is preserved under a natural class of split extensions, which includes certain operations of adding arrows and relations to bound quiver algebras. The paper's contributions include a reduced relative bar resolution, a transparent 'nearly exact' criterion producing long exact sequences in prescribed degrees, and a Jacobi-Zariski sequence for split bounded extensions. The hypotheses in Definitions 3.4 and 3.6 are not decorative: they are exactly what makes the relative resolution finite and the spectral sequence argument work, and the authors are explicit that the closure result is for bounded extensions. The proofs are detailed and mostly self-contained, and the examples in Section 2 illustrate the scope. One step in the proof of Proposition 4.2(1) is incomplete for the case where the bimodule M has projective dimension zero; this is a local gap that is repairable without changing the main theorem.","major_comments":[{"comment":"The proof asserts that H_*(A,A) and H_*(B,B) are isomorphic for * ≥ nu+1. This is justified by Corollary 2.5 (vanishing of H_*(A|B,A) for * ≥ n) together with the Jacobi-Zariski sequence of Theorem 3.8. However, to identify H_*(B,A) with H_*(A,A) at a degree * via the Jacobi-Zariski sequence one needs both H_*(A|B,A)=0 and H_{*+1}(A|B,A)=0. When u=0 and n>1, the claimed range * ≥ nu+1 = 1 is not covered by Corollary 2.5, so the isomorphism is not proved in this case. Either a vanishing statement for H_*(A|B,A) in the low-degree range must be established under the u=0 hypothesis, or the threshold should be replaced by max(nu+1,n); the latter is enough for the main theorem, since Han's conjecture only concerns vanishing in all sufficiently large degrees. In the same paragraph, 'if ∗ ≥ u then H_*(B,M)=0' should read 'if ∗ > u', because Tor_u(B,M) need not vanish when the projective dimension of M is exactly u.","section":"§4, Proposition 4.2(1)"}],"minor_comments":[{"comment":"The displayed sequence 0 → C_*(B,X) → C_*(A,X) → C^M_*(A|B,X) → 0 is not a chain complex in degree 0, since κ_0 ι_0 is the nonzero canonical map X → X_B. Please clarify that the nearly exact structure and the associated double complex are to be formed from the positive-degree part and state how degree 0 is treated separately.","section":"§2, Proposition 2.6"},{"comment":"The Tor arguments are swapped: Proposition 3.3 uses Tor^{Be}_{p+q}(X, M^{⊗_B p}), while Proposition 3.7 writes Tor^{Be}_{p+q}(M^{⊗_B p}, X). If a symmetry of Tor for bimodules is being used, it should be stated explicitly; otherwise the notation should be made consistent.","section":"§3, Propositions 3.3 and 3.7"},{"comment":"In the proof, 'F^1_{p,q} = 0 for p ≥ v' should read 'for p ≥ n', since it is the nilpotency index n that makes M^{⊗_B p} vanish.","section":"§3, Proposition 3.7"},{"comment":"The displayed sequence contains the typo 'Hnu+1m(B, X)'; this should be H_{nu+1}(B, X).","section":"§3, Theorem 3.8"},{"comment":"In the final display, the term A ⊗_B M^{⊗_B n} ⊗_B X is zero when n is the nilpotency index; the finite resolution should be indexed so that the first nonzero term is A ⊗_B M^{⊗_B (n-1)} ⊗_B X, or the indexing of the nilpotency index should be clarified.","section":"§4, Proposition 4.2(2)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious contribution to the literature on Han's conjecture, with a useful new closure result and a clean homological toolbox. The only substantive issue I found is the gap in Proposition 4.2(1) for the u=0 case, which is local and fixable; I therefore recommend major revision rather than rejection. The paper's scope and citation practice are appropriate, and the limitations of the boundedness hypotheses are honestly stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real, if incremental, step on Han's conjecture. The authors prove that if A = B + M is a split bounded extension of finite dimensional algebras, then A satisfies Han's conjecture iff B does. The proof goes through a Jacobi-Zariski long exact sequence for Hochschild homology in large degrees, obtained from a reduced relative bar resolution plus an m-nearly exact sequence criterion. That sequence is new and should be reusable.\n\nThe paper is honest about its scope: the boundedness hypotheses (B-tensor nilpotency, finite projective dimension as a bimodule, and one-sided projectivity) are exactly what make the argument work, and the authors do not claim more. The main theorem gives new examples and connects to arrow-addition/deletion for quiver algebras. The technique is standard homological machinery, applied carefully.\n\nWhat is genuinely new: the closure result itself goes beyond the null-square case in the authors' earlier work, and the Jacobi-Zariski sequence under these hypotheses differs from Kaygun's (which needs B-flatness). The reduced relative bar resolution in Section 2 is a nice generalization of the authors' 1990 lemma.\n\nSoft spots are mostly cosmetic. A few proofs say 'it is easily proven' or 'does not raise any difficulty' at places where the reader has to do work; that is normal for this area but makes the paper denser than necessary. There are also small typographical slips: in Proposition 3.7 the Tor arguments are swapped relative to Proposition 3.3, and the variable for projective dimension shifts from u to v mid-statement. These do not affect the argument: the intended vanishing follows from the bound on projective dimension of M^{⊗ p}. Also, the phrase 'does not raise any difficulty' hides a routine but not totally instant verification of the complex map in Proposition 2.6; I checked enough to believe it.\n\nOne conceptual point worth flagging: the equivalence of vanishing of Hochschild homology in high degrees between A and B is shown via H_*(B,A) = H_*(B,B) for large degrees, using the projective dimension bound. That step is correct. The second part, smoothness equivalence, uses the reduced bar resolution with the contracting homotopy being a right A-module map; that is why the resolution stays exact after applying - ⊗_A X. So the two half-implications fit together properly.\n\nFor a reading group on Hochschild homology and Han's conjecture, this is a good paper: the main theorem is clear, the machinery is instructive, and the limitations are honest. It deserves a serious referee. I would send it to review.\n\nRecommendation: accept after minor fixes (typos and possibly a few expanded 'easily proven' verifications).","headline":"Solid closure theorem for Han's conjecture under split bounded extensions; worth refereeing despite minor typographical slips.","tokens_in":12400,"tokens_out":2480,"would_cite":true,"duration_ms":24561,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G25","16E40","16E30","18G15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Theorem: a finite-dimensional split bounded extension $A = B \\oplus M$ satisfies Han's conjecture if and only if $B$ does.","keywords":["Hochschild homology","Han's conjecture","split bounded extension","global dimension","relative homological algebra","Jacobi-Zariski long exact sequence","quiver algebras","nilpotent bimodule"],"falsifier":"Take any explicitly presented finite-dimensional split bounded extension $A = B \\oplus M$, compute the Hochschild homology groups of $A$ and $B$ in degree $nu+1$ (where $n$ is the $B$-tensor nilpotency index of $M$ and $u$ is the projective dimension of $M$ as a $B$-bimodule), and check whether they are isomorphic. The theorem predicts they are; a single example where they differ would refute the central claim.","tokens_in":11463,"feed_emoji":"","tokens_out":8326,"duration_ms":82076,"temperature":0.7,"pith_summary":"Han's conjecture says that a finite-dimensional algebra has finite global dimension precisely when its Hochschild homology vanishes in all sufficiently high degrees. This paper proves that the class of finite-dimensional algebras satisfying Han's conjecture is closed under split bounded extensions: if $A = B \\oplus M$ with $M$ a bounded $B$-bimodule ideal, then $A$ verifies the conjecture exactly when $B$ does. The result matters because many operations on quiver algebras, such as adding or deleting arrows or certain relations, are special cases of split bounded extensions, so the conjecture can be checked on the smaller base algebra. The proof supplies a Jacobi-Zariski long exact sequence and shows that in high degrees the Hochschild homology groups of $A$ and $B$ are actually isomorphic.","feed_headline":"Split bounded extensions preserve Han's conjecture","feed_subtitle":"For A = B ⊕ M with bounded M, A satisfies Han's conjecture exactly when B does.","key_machinery":"The load-bearing objects are split bounded extensions and the relative homological machinery built around them. A split extension $A = B \\oplus M$ has $M$ as a two-sided ideal and a retraction $A \\to B$; $M$ is bounded when it is $B$-tensor nilpotent (some $n$-fold tensor power $M^{\\otimes_B n}$ vanishes), has finite projective dimension as a $B$-bimodule, and is projective as a left or right $B$-module. The argument runs through a reduced relative bar resolution for $A$ over $B$, a nearly exact sequence of chain complexes whose nearly exactness is proved using the boundedness hypotheses, and then a Jacobi-Zariski long exact sequence for Hochschild homology in large degrees. This sequence, together with a resolution argument, yields both the high-degree isomorphism of Hochschild homologies and the equivalence of smoothness.","core_discovery":"The central claim is Theorem 4.1: for a finite-dimensional split bounded extension $A = B \\oplus M$, the algebra $A$ lies in the class $\\mathcal{H}$ of algebras verifying Han's conjecture if and only if $B$ does. Along the way the paper establishes two sharper facts. First, the Hochschild homology groups of $A$ and $B$ are isomorphic in degrees at least $nu+1$, where $n$ is the $B$-tensor nilpotency index of $M$ and $u$ is its projective dimension as a $B$-bimodule. Second, $A$ is smooth, meaning of finite global dimension, exactly when $B$ is smooth. Thus Han's conjecture is not merely stable under split bounded extensions: the two algebras have identical homological behaviour above an explicit threshold.","pith_inferences":["Because the proof only needs the $B$-bimodule structure of $M$, the same conclusion should hold for any split bounded extension with the same underlying bimodule, regardless of the multiplication on $M$; one could test this by varying the associative product on a fixed bounded bimodule.","The threshold $nu+1$ may not be optimal; computing the actual vanishing degree in explicit examples would test whether the cutoff can be improved.","If a counterexample to Han's conjecture is ever found, this theorem implies it cannot be decomposed as a split bounded extension of a smaller algebra known to satisfy the conjecture, which narrows where counterexamples could hide."],"forward_implications":["For any split bounded extension, verifying Han's conjecture for $A$ is exactly as hard as verifying it for $B$; known cases of $B$ immediately yield new cases of $A$.","In degrees at least $nu+1$, the Hochschild homology groups of $A$ and $B$ are isomorphic, so high-degree vanishing transfers in both directions with an explicit cutoff.","Quiver operations that add or delete arrows, and certain relation changes, when they form a split bounded extension, do not change the Han's conjecture status of the bound quiver algebra.","Finite global dimension is preserved by split bounded extensions, so any construction of this form that is smooth on one side is smooth on the other.","The result does not depend on the associative multiplication on $M$, only on its $B$-bimodule structure, so many non-isomorphic extensions sharing the same underlying bimodule have the same Han's conjecture status."],"supporting_citations":[{"why":"States Han's conjecture, the property whose class is shown to be closed under split bounded extensions.","marker":"[20]"},{"why":"Introduces relative homological algebra and the relative Hochschild homology used throughout the paper.","marker":"[22]"},{"why":"Provides the reduced bar resolution that Theorem 2.3 generalises to split extensions.","marker":"[11]"},{"why":"Supplies the projective resolution of a bimodule used to compute the Tor terms in Proposition 3.3.","marker":"[12]"},{"why":"Gives the bound on projective dimension of tensor powers used to prove that the sequence is nearly exact.","marker":"[10]"}],"fun_headline_variants":["Han's conjecture survives split bounded extensions","Split bounded extensions: Han's conjecture equivalence","If B satisfies Han's, so does A (split bounded)","Han's conjecture stable under split bounded extensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem applies only when the ideal $M$ is bounded in a specific homological sense: its tensor powers over $B$ eventually vanish, it has finite homological size as a $B$-bimodule, and it is projective on at least one side; if any of these conditions fails, the connecting long exact sequence is no longer guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Han's conjecture survives split bounded extensions","Split bounded extensions: Han's conjecture equivalence","If B satisfies Han's, so does A (split bounded)","Han's conjecture stable under split bounded extensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000392,"raw_usage":{"total_tokens":1940,"prompt_tokens":701,"completion_tokens":1239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":317,"completion_tokens_details":{"reasoning_tokens":1190}},"tokens_in":317,"tokens_out":1239,"duration_ms":10724,"temperature":1.0,"reasoning_tokens":1190,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:23:08.038638+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any explicitly presented finite-dimensional split bounded extension $A = B \\oplus M$, compute the Hochschild homology groups of $A$ and $B$ in degree $nu+1$ (where $n$ is the $B$-tensor nilpotency index of $M$ and $u$ is the projective dimension of $M$ as a $B$-bimodule), and check whether they are isomorphic. The theorem predicts they are; a single example where they differ would refute the central claim.","supporting_citations":[{"cited_title":"Hochschild (co)homology dimension","cited_arxiv_id":null,"evidence_quote":"States Han's conjecture, the property whose class is shown to be closed under split bounded extensions."},{"cited_title":"Relative homological algebra, Trans","cited_arxiv_id":null,"evidence_quote":"Introduces relative homological algebra and the relative Hochschild homology used throughout the paper."},{"cited_title":"Rigidity of truncated quiver algebras","cited_arxiv_id":null,"evidence_quote":"Provides the reduced bar resolution that Theorem 2.3 generalises to split extensions."},{"cited_title":"Tensor Hochschild homology and cohomology","cited_arxiv_id":null,"evidence_quote":"Supplies the projective resolution of a bimodule used to compute the Tor terms in Proposition 3.3."},{"cited_title":"Homological algebra","cited_arxiv_id":null,"evidence_quote":"Gives the bound on projective dimension of tensor powers used to prove that the sequence is nearly exact."}],"review_version":1}