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Han's Conjecture

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a-counterexample-to-han-s-conjectureAlgebraposed by Yang Han, 2006recorded: disproved

1 attempt · no person has looked

Statement

For a finite-dimensional algebra AA, finite global dimension forces HHn(A)=0\mathrm{HH}_n(A) = 0 for all large nn. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional C\mathbb{C}-algebra with HHn(A)=0\mathrm{HH}_n(A) = 0 for every n≥1n \geq 1 and gldim A=∞\mathrm{gldim}\, A = \infty, built by transporting Krah's phantom into a singularity category via one-periodic folding.

Context

The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.

A named conjecture of the representation theory of finite-dimensional algebras, open for twenty years, with its own survey article, and the surviving half of the Happel-Han pair after the cohomology version fell in 2005.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.6 Sol Ultra with Bochao Kong, Yeqin Liu, Yu Shen ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol Ultra
    people
    Bochao Kong, Yeqin Liu, Yu Shen

    The paper's acknowledgment in full: "The counterexample presented in this paper was discovered with the assistance of OpenAI's GPT-5.6 Sol Ultra model. All mathematical arguments and references were independently verified by the authors." In a counterexample paper the algebra is the whole result, so crediting the model with its discovery is a claim about the central object, not about support work. The hedge "with the assistance of" keeps this below the top tier.

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