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Keller’s strong homological conjecture

For every dg algebra AA over a field kk such that dim⁡kH∗(A)<∞\dim_k H^*(A)<\infty, the regular module AA cogenerates the unbounded derived category D(A)D(A) of dg AA-modules; equivalently, for every M∈D(A)M\in D(A), if Hom⁡D(A)(M,A[n])=0\operatorname{Hom}_{D(A)}(M,A[n])=0 for all n∈Zn\in\mathbb{Z}, then M≃0M\simeq 0.

References

Primary source

arXiv

Additional references

derived-categorieshomological-algebrasymplectic-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed manuscript claims a counterexample, so the conjecture may be false, but the claim has not been independently verified.

Keller’s strong homological conjecture predicts that a finite-dimensional cohomology condition forces a corresponding cogeneration property for an unbounded derived category. A September 2026 manuscript claims to construct a coconnective dg algebra violating that property.

September 2026 counterexample claim

Mingyuan Hu, Vivek Shende, and Dinglong Wang present a coconnective dg algebra with finite-dimensional cohomology whose module category is not cogenerated as conjectured. The manuscript states that all examples and most proofs were produced by ChatGPT; it is unrefereed, so the claimed disproof remains unconfirmed.

Current status (as of September 2026): The conjecture has a claimed counterexample, but its correctness and the resulting disproof remain unverified.

  • ChatGPTOpenAIsolved2026-09-28evidence

    A dg-algebra counterexample challenges Keller’s strong homological conjecture

Sources

Solutions 0

No solutions have been posted yet.