Whether periodic simple modules force algebra periodicity
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If all simple modules of a finite-dimensional algebra are periodic, must the algebra itself be periodic? The focused variant also assumes twisted periodicity.
Why it matters and what counts as progress
Why it matters: The implication would pass periodicity from the simplest representations to the full algebra.
What counts as progress: A general implication or a twisted-periodic algebra with periodic simples and a rigorously nonperiodic module, showing that ordinary algebra periodicity fails.
How the problem developed
- 9 Sept 2026 · Counterexample [1]
Haruhisa Enomoto Constructed a 36-dimensional algebra whose simple modules all have period four but which has a two-dimensional nonperiodic module, so the algebra itself is not periodic.
AI contribution
A counterexample to the periodicity conjecture for finite-dimensional algebras
Enomoto constructs a 36-dimensional algebra Λ(q) over an algebraically closed field with an element q of infinite multiplicative order. Every simple right module has period four and the algebra is twisted periodic, yet a two-dimensional module has pairwise nonisomorphic even syzygies, so the algebra is not periodic.
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