Let A be a finite-dimensional algebra over a field. Assume that A satisfies the Auslander condition: for every finitely generated left A-module M, there exists an integer b_M such that operatornameExt^i_A(M,A)=0 for all i>b_M. Then A is Iwanaga–Gorenstein; equivalently, operatornameid_A A<∞ and operatornameid_A^opA<∞.

Auslander–Gorenstein conjecture

The conjecture asks whether an algebra satisfying the Auslander condition must be Gorenstein. Auslander and Reiten formulated the question; it remains open for general finite-dimensional algebras.

Known results

Monomial algebras satisfy stronger bounded versions; with $n$ simple modules, $(4n-2)$ -Gorenstein implies Iwanaga–Gorenstein with injective dimension at most $4n-2$ .
Klász (2025) proved stronger results for monomial algebras, including that every $2n$ -Gorenstein monomial algebra is $(2n+1)$ -Gorenstein.
Finite finitistic dimension, and certain grade-condition hypotheses, imply the conjecture; the relevant general finitistic-dimension problem remains open.
Frobenius and stable-equivalence results transfer the conjecture between suitable algebras but do not prove it generally.

September 2026 Frobenius-extension results

Zhibing Zhao's preprint gives conditional transfer results under suitable Frobenius extensions and constructs non-Gorenstein algebras with stabilized torsionfree filtrations. It is claimed progress on the structure of the problem, not a proof or counterexample for the general conjecture.

Current status (as of September 2026): The conjecture remains open in general; substantial restricted and transfer results are known, while Zhao's latest claimed advance is unverified and does not resolve it.

Auslander–Gorenstein conjecture

Primary source

Additional references

Progress summary

Solutions 0