The Second Finitistic Dimension Conjecture asserts that the little (small) finitistic dimension of every Artin algebra is finite: the supremum of the projective dimensions of all finitely generated modules of finite projective dimension over a fixed algebra never reaches infinity. Originating in problems publicized by Bass in 1960, it is the longest-standing open homological conjecture for Artin algebras and is what most authors simply call the Finitistic Dimension Conjecture. The conjecture asserts finiteness only, with no uniform bound in the number of isomorphism classes of simple modules; it remains open in general.

Background

An Artin algebra is a ring that is artinian on both sides and finitely generated as a module over its center; the leading examples are finite-dimensional algebras over a field, presented concretely as kQ/IkQ/I for a finite quiver QQ (a directed graph) and an ideal II of relations in the path algebra kQkQ (such a presentation is called a bound quiver algebra). All modules below are understood to be finitely generated, which for an Artin algebra is equivalent to having finite length. The projective dimension pd⁡ΛM\operatorname{pd}_\Lambda M of a module MM is the minimal length of a projective resolution of MM (possibly infinite). In his 1960 study of homological generalizations of semi-primary rings, Bass introduced the little (small) finitistic dimension

fpd⁡(Λ)=sup⁡{pd⁡ΛM∣M∈mod⁡Λ, pd⁡ΛM<∞}\operatorname{fpd}(\Lambda)=\sup\{\operatorname{pd}_\Lambda M \mid M\in\operatorname{mod}\Lambda,\ \operatorname{pd}_\Lambda M<\infty\}

(also written findim⁡Λ\operatorname{findim}\Lambda) and its big analogue FPD⁡(Λ)\operatorname{FPD}(\Lambda), where the supremum runs over all, not necessarily finitely generated, modules of finite projective dimension. In the standard numbering recalled by Huisgen-Zimmermann's survey, the Finitistic Dimension Conjectures for finite-dimensional algebras are: (I) fpd⁡(Λ)=FPD⁡(Λ)\operatorname{fpd}(\Lambda)=\operatorname{FPD}(\Lambda), and (II) fpd⁡(Λ)<∞\operatorname{fpd}(\Lambda)<\infty. Conjecture (II) is the Second Finitistic Dimension Conjecture, and is what most later authors call, simply, the Finitistic Dimension Conjecture. The conjecture is stated in the literature both for finite-dimensional algebras over a field (as in the survey just quoted) and for Artin algebras (as by Moradifar–Šaroch, who discuss its validity over an Artin algebra); since every finite-dimensional algebra over a field is an Artin algebra, the Artin-algebra form is the broader one, and no reduction of the Artin case to the finite-dimensional case is assumed. Conjecture (I) is false: Huisgen-Zimmermann (1992) constructed monomial-relation algebras (algebras kQ/IkQ/I with II generated by paths) with fpd⁡(Λ)=n\operatorname{fpd}(\Lambda)=n and FPD⁡(Λ)=n+1\operatorname{FPD}(\Lambda)=n+1 for every n≥2n\ge 2, and Smalø (2000) showed the gap can be arbitrarily large. Conjecture (II) is verified for many classes — algebras of finite representation type (finitely many indecomposable modules up to isomorphism), monomial-relation algebras, algebras whose Jacobson radical JJ (the intersection of all maximal left ideals) satisfies J2=0J^2=0 or J3=0J^3=0, and algebras of quasi-uniform Loewy length (a condition on the Loewy lengths, i.e. the number of steps in the radical filtration, of the indecomposable projective modules) — but is open in general. Three lines frame the current frontier. First, approximation theory: Auslander and Reiten (1991) proved the conjecture follows from contravariant finiteness of the category of finitely generated modules of finite projective dimension (every module admits a right approximation by such modules); Igusa–Smalø–Todorov (1990) showed this sufficient condition is not necessary, so it cannot settle the general case, and Moradifar–Šaroch (2022) proved the analogous condition on the category of modules of finite Gorenstein projective dimension (cycles of acyclic complexes of projectives that remain exact after applying Hom into projectives) suffices over left artinian rings. Second, derived categories: Rickard (2019) proved that if the injective modules generate the unbounded derived category of a finite-dimensional algebra as a triangulated category with coproducts — the derived category is the category of unbounded complexes of modules with chain maps inverted up to homotopy, and generation means that every module is obtained from injectives using finite triangles and arbitrary direct sums — then the conjecture holds for that algebra, and Cummings (2023) proved that, for finite-dimensional algebras, the conjecture is equivalent to a left-right symmetry statement for finitistic dimension and for injective generation. Third, quiver reductions: radical preservation (Giatagantzidis 2024), monomial arrow removal (Erdmann–Giatagantzidis–Psaroudakis–Solberg 2025), and general arrow reductions (Giatagantzidis 2025) reduce the conjecture for a given bound quiver algebra to a smaller algebra.

Problem Statement

Let Λ\Lambda be an Artin algebra (in particular, a finite-dimensional algebra over a field, given as a bound quiver algebra kQ/IkQ/I), and let mod⁡Λ\operatorname{mod}\Lambda be its finitely generated modules. Define the little (small) finitistic dimension

fpd⁡(Λ)=sup⁡{pd⁡ΛM∣M∈mod⁡Λ, pd⁡ΛM<∞}.\operatorname{fpd}(\Lambda)=\sup\{\operatorname{pd}_\Lambda M \mid M\in\operatorname{mod}\Lambda,\ \operatorname{pd}_\Lambda M<\infty\}.

The Second Finitistic Dimension Conjecture asserts:

fpd⁡(Λ)<∞ for every Artin algebra Λ.\operatorname{fpd}(\Lambda)<\infty\ \text{for every Artin algebra }\Lambda.

Since every finite-dimensional algebra over a field is an Artin algebra, the conjecture in particular asserts that no finite-dimensional algebra over a field — for instance, no bound quiver algebra kQ/IkQ/I — admits finitely generated modules of arbitrarily large finite projective dimension. The conjecture asserts finiteness only; it makes no claim of a uniform bound in the number of isomorphism classes of simple modules.

References

  1. Hyman Bass, 'Finitistic dimension and a homological generalization of semi-primary rings', Transactions of the American Mathematical Society (1960), DOI: 10.1090/s0002-9947-1960-0157984-8, https://doi.org/10.1090/s0002-9947-1960-0157984-8
  2. Birge Zimmermann Huisgen, 'Homological domino effects and the first Finitistic Dimension Conjecture', Inventiones Mathematicae (1992), DOI: 10.1007/bf02100610, https://doi.org/10.1007/bf02100610
  3. Birge Huisgen-Zimmermann, 'The finitistic dimension conjectures -- a tale of 3.5 decades', in A. Facchini and C. Menini (eds.), Abelian Groups and Modules, Kluwer (1995), pp. 501-517, DOI: 10.1007/978-94-011-0443-2_41, https://doi.org/10.1007/978-94-011-0443-2_41 (survey; arXiv:1407.2383, https://arxiv.org/abs/1407.2383)
  4. Sverre O. Smalø, 'Homological Differences between Finite and Infinite Dimensional Representations of Algebras', in Infinite Length Modules (Birkhäuser, 2000), DOI: 10.1007/978-3-0348-8426-6_23, https://doi.org/10.1007/978-3-0348-8426-6_23
  5. Maurice Auslander, Idun Reiten, 'Applications of contravariantly finite subcategories', Advances in Mathematics (1991), DOI: 10.1016/0001-8708(91)90037-8, https://doi.org/10.1016/0001-8708(91)90037-8
  6. K. Igusa, S. O. Smalø, G. Todorov, 'Finite projectivity and contravariant finiteness', Proceedings of the American Mathematical Society (1990), DOI: 10.1090/s0002-9939-1990-1027094-0, https://doi.org/10.1090/s0002-9939-1990-1027094-0
  7. Jeremy Rickard, 'Unbounded derived categories and the finitistic dimension conjecture', Advances in Mathematics (2019), DOI: 10.1016/j.aim.2019.106735, arXiv:1804.09801, https://arxiv.org/abs/1804.09801
  8. Pooyan Moradifar, Jan Šaroch, 'Finitistic Dimension Conjectures via Gorenstein Projective Dimension', Journal of Algebra (2022), DOI: 10.1016/j.jalgebra.2021.10.026, arXiv:2006.02182, https://arxiv.org/abs/2006.02182
  9. Eric J. Hanson, Kiyoshi Igusa, 'A counterexample to the ϕ\phi-dimension conjecture', Mathematische Zeitschrift 300 (2022), 807-826, DOI: 10.1007/s00209-021-02795-7, https://doi.org/10.1007/s00209-021-02795-7, arXiv:1911.00614, https://arxiv.org/abs/1911.00614
  10. Charley Cummings, 'Left-right symmetry of finite finitistic dimension', Bulletin of the London Mathematical Society (2023), DOI: 10.1112/blms.12954, arXiv:2211.04394, https://arxiv.org/abs/2211.04394
  11. Sergio Estrada, Alina Iacob, 'Gorenstein projective precovers and finitely presented modules', Rocky Mountain Journal of Mathematics 54 (2024), no. 3, DOI: 10.1216/rmj.2024.54.715, https://doi.org/10.1216/rmj.2024.54.715, arXiv:2303.00213, https://arxiv.org/abs/2303.00213
  12. Odysseas Giatagantzidis, 'Radical preservation and the finitistic dimension', Bulletin of the London Mathematical Society 58 (2026), DOI: 10.1112/blms.70222, https://doi.org/10.1112/blms.70222, arXiv:2412.10965, https://arxiv.org/abs/2412.10965
  13. Karin Erdmann, Odysseas Giatagantzidis, Chrysostomos Psaroudakis, Øyvind Solberg, 'Monomial arrow removal and the finitistic dimension conjecture', arXiv:2506.23747 (2025), https://arxiv.org/abs/2506.23747
  14. Odysseas Giatagantzidis, 'Arrow reductions for the finitistic dimension conjecture', arXiv:2507.12978 (2025), https://arxiv.org/abs/2507.12978
  15. Panagiotis Kostas, 'Finitistic dimension via modules over the singularity category', arXiv:2608.15482 (2026), https://arxiv.org/abs/2608.15482
  16. Jan Trlifaj, Mykyta Dubov, 'Tilting modules for the Cummings construction', arXiv:2606.10204 (2026), https://arxiv.org/abs/2606.10204