MathDB

Auslander–Reiten conjecture

For every finite-dimensional algebra AA over a field and every finitely generated AA-module MM, if Ext⁡Ai(M,M)=0\operatorname{Ext}_A^i(M,M)=0 for all i>0i>0 and MM is a generator, equivalently A∈add⁡(M)A\in\operatorname{add}(M), then MM is projective.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Ext-vanishing formulation

    For every finite-dimensional algebra AA and finitely generated AA-module MM, if Ext⁡Ai(M,M⊕A)=0\operatorname{Ext}_A^i(M,M\oplus A)=0 for all i>0i>0, then MM is projective.

    source: The Auslander-Reiten conjecture for algebras with radical cube zero

References

Primary source

arXiv

Additional references

homological-algebrarepresentation-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper handles algebras whose radical has cube zero, but the full conjecture remains open.

The Auslander–Reiten conjecture asks whether self-orthogonal generators in the relevant algebraic settings must be projective. The latest result treats the substantial special class with radical satisfying J3=0J^3=0, not the general conjecture.

Known results

  • May 2024: the conjecture holds for several finitely generated modules over commutative Noetherian rings under finite complete-intersection-dimension hypotheses.
  • January 2026: it holds under finite complete-intersection injective-dimension assumptions and related finite-dimension conditions.
  • March 2026: it holds for centralizer matrix algebras.
  • September 2025: it holds for modules of finite quasi-projective dimension over left Noetherian rings.

September 2026 radical-cube-zero result

A September 2026 preprint claims the conjecture for algebras with J3=0J^3=0, together with an explicit self-extension bound. This is a special-case advance; no source retrieved here claims a proof of the full conjecture.

Current status (as of September 2026): The conjecture is established only in special classes, including J3=0J^3=0 algebras; the general conjecture remains open, and the newest special-case claim is unverified.

Sources

Solutions 0

No solutions have been posted yet.