Math Conjectures

Auslander–Reiten Conjecture

OPENMajorConjectureProposed 1975 · Standard version

Canonical statement

Let AA be an Artin algebra and MM a finitely generated left AA-module. If
Ext⁡Ai(M,M⊕A)=0for every i>0, \operatorname{Ext}^i_A(M,M\oplus A)=0 \quad\text{for every }i>0,
then MM is projective.
View source LaTeX
Let \(A\) be an Artin algebra and \(M\) a finitely generated left \(A\)-module. If
\[
  \operatorname{Ext}^i_A(M,M\oplus A)=0
    \quad\text{for every }i>0,
\] then \(M\) is projective.
ALG-REP-009:[AuslanderReiten1975][Happel1991Homological][PanXi2023ARC]
Algebra and representation theory

Notes

The conjecture asks whether the vanishing of higher extensions can detect projectivity: if MM is a finitely generated module over an Artin algebra AA and Ext⁡Ai(M,M⊕A)=0\operatorname{Ext}^i_A(M, M\oplus A)=0 for every i>0i>0, must MM be projective? It was posed by Auslander and Reiten in 1975, in the same paper in which they formulated their generalized version of the Nakayama conjecture, to which it is closely tied [AuslanderReiten1975].

The problem belongs to the family of homological conjectures for finite-dimensional algebras surveyed by Happel [Happel1991Homological]. It has been confirmed for many well-structured classes, including a range of Gorenstein, symmetric, and representation-finite algebras, and one productive line of attack transfers the property between algebras along equivalences, for instance singular equivalences [PanXi2023ARC].

Despite this, the conjecture is not known even for broad families of self-injective algebras, where the statement is particularly natural. It remains open; a resolution requires either a proof covering arbitrary Artin algebras or a non-projective module with no higher self-extensions and no higher extensions against the algebra.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.