The (second) finitistic dimension conjecture, publicized by Bass in 1960, asks whether the little finitistic dimension of an Artin algebra — the supremum of the projective dimensions of its finitely generated modules of finite projective dimension — is always finite. It is one of the longest-standing open problems in the homological theory of finite-dimensional algebras: it is known for many special classes and for all algebras of representation dimension at most 3 (Igusa–Todorov), yet general finiteness results such as Iyama's theorem that representation dimension is always finite do not settle it, and reduction programs via quasi-hereditary algebras and via arrow removal remain unfinished. The open problem was highlighted, in essentially these terms, by Zhang and Zhang's 2008 paper; the conjecture remains open.

Background

An Artin algebra is a ring AA that is finitely generated as a module over its center, the center being a commutative artinian ring; the leading example is a finite-dimensional algebra over a field. For a finitely generated left AA-module MM, the projective dimension pd⁡AM\operatorname{pd}_A M is the minimal length of a projective resolution of MM (an exact sequence of projective modules ending in MM), and ∞\infty if no finite one exists. The (little) finitistic dimension of AA is

fin.dim⁡A=sup⁡{pd⁡AM∣M finitely generated with pd⁡AM<∞},\operatorname{fin.dim} A=\sup\{\operatorname{pd}_A M \mid M \text{ finitely generated with } \operatorname{pd}_A M<\infty\},

the supremum taken only over modules of finite projective dimension; the big finitistic dimension Fin.dim⁡A\operatorname{Fin.dim} A is the analogous supremum over arbitrary left modules. Bass's 1960 paper publicized two conjectures about these invariants: that fin.dim⁡A=Fin.dim⁡A\operatorname{fin.dim} A=\operatorname{Fin.dim} A, and that fin.dim⁡A<∞\operatorname{fin.dim} A<\infty. The first fails in general (the survey of Huisgen-Zimmermann documents monomial-relation examples with Fin.dim⁡A=fin.dim⁡A+1\operatorname{Fin.dim} A=\operatorname{fin.dim} A+1); the second — the finitistic dimension conjecture — is open and is among the oldest unsolved problems in the representation theory of algebras.

The conjecture is known for many classes: monomial-relation algebras (algebras presented as kQ/IkQ/I, where QQ is a quiver — a directed graph whose directed paths multiply — and II is an ideal generated by paths), algebras whose Jacobson radical JJ (the intersection of the maximal left ideals) satisfies J3=0J^3=0, algebras whose category of finite-projective-dimension modules admits good approximations, and others surveyed by Huisgen-Zimmermann. Two general frameworks dominate the modern attacks. First, Auslander's representation dimension: rep.dim⁡A\operatorname{rep.dim} A is the infimum of the global dimensions (suprema of projective dimensions over all modules) of endomorphism rings End⁡A(V)\operatorname{End}_A(V) as VV ranges over generator-cogenerators, that is, modules containing every indecomposable projective and injective as a direct summand. Igusa and Todorov proved that rep.dim⁡A≤3\operatorname{rep.dim} A\le 3 implies fin.dim⁡A<∞\operatorname{fin.dim} A<\infty, which made representation dimension the main sufficient route: large classes are known to have rep.dim⁡≤3\operatorname{rep.dim}\le 3, for instance algebras radically embedded in representation-finite algebras — a radical embedding being an inclusion of algebras A⊂BA\subset B under which the radical of AA is an ideal of BB, and representation-finite meaning that, up to isomorphism, only finitely many indecomposable finitely generated modules exist — (Erdmann–Holm–Iyama–Schröer), and torsionless-finite classes, meaning classes with only finitely many indecomposable modules embedding into projective modules (Assem–Coelho–Trepode). Iyama proved that rep.dim⁡A<∞\operatorname{rep.dim} A<\infty for every Artin algebra — but Rouquier constructed exterior algebras of representation dimension greater than 3, and representation dimension is unbounded, so Iyama's finiteness is strictly weaker than what the Igusa–Todorov criterion needs and does not settle the conjecture. Second, reduction to quasi-hereditary algebras: a quasi-hereditary algebra is one carrying a chain of idempotent ideals (a heredity chain) with respect to which its standard modules (certain quotients of the indecomposable projective modules determined by the chain) form a well-behaved stratification, and every Artin algebra is an endomorphism algebra of a projective-injective module over a quasi-hereditary algebra. Xi and Wei transferred finiteness of the finitistic dimension along left idealized extensions (inclusions of algebras A⊆RA\subseteq R sharing the same identity element, in which the radical of AA is a left ideal of RR), which reduces the conjecture to whether every quasi-hereditary algebra admits such an extension that is monomial or has representation dimension at most 3. Closely tied to the conjecture is the Nakayama conjecture: if every term of the minimal injective resolution of AA{}_AA (the algebra viewed as a left module over itself) is projective (equivalently, AA has infinite dominant dimension, the dominant dimension being the number of leading projective terms of that resolution), then AA is self-injective, i.e. injective as a left module over itself; it is a classical consequence of the finitistic dimension conjecture and is likewise open.

Problem Statement

Let AA be an Artin algebra and let

fin.dim⁡A=sup⁡{pd⁡AM∣M a finitely generated left A-module with pd⁡AM<∞}.\operatorname{fin.dim} A=\sup\{\operatorname{pd}_A M \mid M \text{ a finitely generated left } A\text{-module with } \operatorname{pd}_A M<\infty\}.

Is fin.dim⁡A<∞\operatorname{fin.dim} A<\infty for every Artin algebra AA (the central special case being finite-dimensional algebras over a field)? A complete answer must settle the question for the entire class: either a proof that the supremum is finite for every Artin algebra, or an explicit Artin algebra together with finitely generated modules of arbitrarily large finite projective dimension. No restriction on the class of algebras (ground ring, number of simple modules, Loewy length — the nilpotency index of the radical — representation type, dimension) is permitted, and the conjecture may not be replaced by its restriction to any proper subclass.

References

  1. Aiping Zhang, Shunhua Zhang, On the finitistic dimension conjecture of Artin algebras, Journal of Algebra 320 (2008), no. 1, 253-258. DOI: 10.1016/j.jalgebra.2007.12.011. https://doi.org/10.1016/j.jalgebra.2007.12.011
  2. H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Transactions of the American Mathematical Society 95 (1960), 466-488. DOI: 10.1090/S0002-9947-1960-0157984-8. https://doi.org/10.1090/S0002-9947-1960-0157984-8
  3. K. Igusa, G. Todorov, On the finitistic global dimension conjecture for Artin algebras, in: Representations of Algebras and Related Topics, Fields Institute Communications 45, American Mathematical Society (2005), 201-204. DOI: 10.1090/fic/045/15. https://doi.org/10.1090/fic/045/15
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  7. Birge Huisgen-Zimmermann, The finitistic dimension conjectures - a tale of 3.5 decades, arXiv:1407.2383 (2014). https://arxiv.org/abs/1407.2383
  8. C. Xi, On the finitistic dimension conjecture II: related to finite global dimension, Advances in Mathematics 201 (2006), 116-142. DOI: 10.1016/j.aim.2005.02.002. https://doi.org/10.1016/j.aim.2005.02.002
  9. Jiaqun Wei, Finitistic and Representation Dimensions, arXiv:0803.3364 (2008). https://arxiv.org/abs/0803.3364
  10. Claus Michael Ringel, Iyama's finiteness theorem via strongly quasi-hereditary algebras, arXiv:0912.5001 (2009). https://arxiv.org/abs/0912.5001
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  12. Liran Shaul, The finitistic dimension conjecture via DG-rings, arXiv:2209.02068 (2022). https://arxiv.org/abs/2209.02068
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  14. Odysseas Giatagantzidis, Arrow reductions for the finitistic dimension conjecture, arXiv:2507.12978 (2025). https://arxiv.org/abs/2507.12978