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Split bounded extension algebras and Han's conjecture

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Pith's one-line read Theorem: a finite-dimensional split bounded extension $A = B \oplus M$ satisfies Han's conjecture if and only if $B$ does.

desk verdict Solid closure theorem for Han's conjecture under split bounded extensions; worth refereeing despite minor typographical slips. read the letter →

arxiv 1908.11130 v3 pith:D5XDE3HI submitted 2019-08-29 math.KT math.RAmath.RT

Claude Cibils , Marcelo Lanzilotta , Eduardo N. Marcos , Andrea Solotar This is my paper · ORCID
classification math.KTmath.RAmath.RT MSC 18G2516E4016E3018G15
keywords HochschildhomologyHan'sconjecturesplitboundedextensionglobaldimensionrelativehomologicalalgebraJacobi-Zariskilongexactsequencequiveralgebrasnilpotentbimodule
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

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

Referee Report

1 major / 5 minor

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.

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 (1)
  1. [§4, Proposition 4.2(1)] 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.
minor comments (5)
  1. [§2, Proposition 2.6] 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.
  2. [§3, Propositions 3.3 and 3.7] 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.
  3. [§3, Proposition 3.7] 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.
  4. [§3, Theorem 3.8] The displayed sequence contains the typo 'Hnu+1m(B, X)'; this should be H_{nu+1}(B, X).
  5. [§4, Proposition 4.2(2)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from the definitions and standard homological algebra, not from Han's conjecture.

full rationale

The derivation chain is self-contained. Theorem 4.1 is obtained from Proposition 4.2(1) and (2). Part (1) uses Corollary 2.5, which follows from the reduced relative bar resolution proved in Theorem 2.3, and the Jacobi-Zariski long exact sequence of Theorem 3.8, which is derived via m-nearly exact sequences and a spectral sequence argument; none of these steps assumes the vanishing of Hochschild homology that is being proved. Part (2) uses standard facts about projective modules together with the finite reduced relative bar resolution whose contracting homotopy is verified in the paper. The boundedness hypotheses in Definitions 3.4 and 3.6 are explicit assumptions, not renamed conclusions. The one author-overlapping result invoked as a tool, Proposition 4.1 of [12] in the proof of Proposition 3.3, is a concrete projective resolution of M as a Be-module; it is parameter-free, has stated assumptions that do not include Han's conjecture, and is not equivalent to Theorem 4.1. Self-citations such as [11], [13], and [14] are contextual or technical, and the central claim does not reduce to them. The typographical issues noted in Proposition 3.7 do not affect the mathematical content. Therefore no circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. Its assumptions are the standard framework of homological algebra plus the boundedness hypotheses defining split bounded extensions.

assumptions (4)
  • standard math Standard results in homological algebra: spectral sequence convergence, derived functors, and Cartan-Eilenberg Chapter IX Proposition 2.6 on projective dimensions.
    Used in Section 3 to derive the projective dimension bound for M⊗_B p and to set up the spectral sequences.
  • standard math The relatively projective resolution of a B-bimodule M provided in Cibils [12] computes Tor^{Be}_*.
    Invoked in the proof of Proposition 3.3 to identify the homology of column p=1 as Tor^{Be}_*(X,M).
  • domain assumption The direct implication of Han's conjecture: finite global dimension implies Hochschild homology vanishes in large degrees.
    Cited in the introduction as true; it is background context, not used in the proof of Theorem 4.1.
  • domain assumption The boundedness hypotheses defining a split bounded extension: M is B-tensor nilpotent, of finite projective dimension as a B-bimodule, and projective as a left or right B-module.
    This is the definition (3.4, 3.6) of the class of extensions to which the theorem applies; the proof of the Jacobi-Zariski sequence and the smoothness transfer both rely on these properties.

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Pith. "Pith review of Split bounded extension algebras and Han's conjecture." pith.science (2026). https://pith.science/paper/D5XDE3HI

@misc{pith2026190811130,
  author       = {Pith},
  title        = {Pith review of: Split bounded extension algebras and Han's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5XDE3HI}},
  note         = {Machine review of arXiv:1908.11130}
}
read the original abstract

A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions.

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Cited by 1 Pith paper

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