Finitistic dimension conjecture for Artin algebras

Establish that every Artin algebra has finite finitistic dimension, equivalently prove the finitistic dimension conjecture for Artin algebras.

Background

The paper frames its categorical study around the classical finitistic dimension conjecture. For an algebra, the finitistic dimension is the supremum of the projective dimensions of finitely generated modules that have finite projective dimension. The conjecture asserts that this supremum is finite for every Artin algebra. The authors identify this as a longstanding unresolved problem and note that it implies several other homological conjectures, including the Auslander–Reiten, Nakayama, and Wakamatsu tilting conjectures.

References

A longstanding open problem is the finitistic dimension conjecture, which asserts that every Artin algebra has finite finitistic dimension , where the finitistic dimension $\operatorname{fpd}\Lambda$ of an algebra $\Lambda$ is defined as the supremum of the projective dimensions of all finitely generated modules having finite projective dimension. This conjecture implies several other homological conjectures, including the Auslander-Reiten conjecture, the Nakayama conjecture, the Wakamatsu tilting conjecture. Despite recent progress on finitistic dimensions (), the finitistic dimension conjecture remains open.

— Finitistic dimensions in triangulated categories with a compact silting generator  (2608.26541 - Yang, 27 Aug 2026) in Section 1, Introduction