Bravo and Paquette (2020) proved that if Λ\Lambda is an Artin algebra and ee is a primitive idempotent whose associated semisimple module SeS_e has finite projective dimension, then findim⁡((1−e)Λ(1−e))≤2findim⁡(Λ)−ℓ\operatorname{findim}((1-e)\Lambda(1-e))\le 2\operatorname{findim}(\Lambda)-\ell, where ℓ\ell is determined by the last nonvanishing self-extension group of SeS_e. By a classical construction of Auslander, generalized by Iyama, every Artin algebra arises as such a corner algebra (1−e)Λ(1−e)(1-e)\Lambda(1-e) of an algebra Λ\Lambda of finite global dimension, so the authors explicitly state that extending their theorem to non-primitive idempotents would yield a complete proof of the finitistic dimension conjecture. The problem posed here is exactly that extension: prove that findim⁡((1−e)Λ(1−e))\operatorname{findim}((1-e)\Lambda(1-e)) is finite under the same hypotheses when ee is an arbitrary idempotent. Later literature (Klapproth's minimal resolutions over idempotent subrings, arrow-removal and bifinite-extension reduction techniques through 2025) develops adjacent machinery but leaves this question open.

Background

An Artin algebra is a ring that is finitely generated as a module over its center, which is assumed to be Artinian; the basic example is a finite-dimensional algebra over a field, presented as a quotient kQ/IkQ/I of a path algebra kQkQ of a finite quiver (directed graph) QQ by an admissible ideal II. For a finitely generated left module MM over such an algebra Λ\Lambda, the projective dimension pd⁡ΛM\operatorname{pd}_\Lambda M is the length of the shortest projective resolution of MM (possibly infinite). The (little) finitistic dimension of Λ\Lambda is findim⁡Λ=sup⁡{pd⁡ΛM:M finitely generated with pd⁡ΛM<∞}\operatorname{findim}\Lambda=\sup\{\operatorname{pd}_\Lambda M : M \text{ finitely generated with } \operatorname{pd}_\Lambda M<\infty\}. Bass publicized in 1960 the conjecture that findim⁡Λ\operatorname{findim}\Lambda is finite for every Artin algebra; this finitistic dimension conjecture (FDC) remains one of the major open homological conjectures in the representation theory of algebras, and a positive answer would imply several other classical conjectures, including the Nakayama and generalized Nakayama conjectures, the Auslander–Reiten conjecture, and the Gorenstein symmetry conjecture — a family of long-standing homological vanishing conjectures for Artin algebras.

A standard strategy for attacking the FDC is to compare the homological properties of Λ\Lambda with those of a corner algebra: for an idempotent ee (e2=ee^2=e) of Λ\Lambda, one forms the corner algebra Γ=(1−e)Λ(1−e)\Gamma=(1-e)\Lambda(1-e), which is naturally the endomorphism algebra of the projective right Λ\Lambda-module (1−e)Λ(1-e)\Lambda. An idempotent is primitive if it cannot be written as a sum of two nonzero idempotents, equivalently if the semisimple module Se=Λe/rad⁡ΛeS_e=\Lambda e/\operatorname{rad}\Lambda e — where rad⁡\operatorname{rad} denotes the Jacobson radical — is simple. Earlier comparisons of findim⁡Λ\operatorname{findim}\Lambda and corner algebras include work of Fuller and Saorín (1992) and Xi (2008), and the arrow-removal reduction technique of Green–Psaroudakis–Solberg (published 2021), which Bravo–Paquette's theorem recovers as an application. A key structural fact is a construction of Auslander (1971), later generalized by Iyama (2002): every Artin algebra Γ\Gamma can be realized as Γ=(1−e)Λ(1−e)\Gamma=(1-e)\Lambda(1-e) for some idempotent ee of an Artin algebra Λ\Lambda with finite global dimension (finite global dimension means every module has finite projective dimension, so in particular findim⁡Λ<∞\operatorname{findim}\Lambda<\infty and pd⁡ΛSe<∞\operatorname{pd}_\Lambda S_e<\infty).

Bravo and Paquette proved: if ee is primitive, findim⁡Λ=r<∞\operatorname{findim}\Lambda=r<\infty, and pd⁡ΛSe<∞\operatorname{pd}_\Lambda S_e<\infty, then findim⁡Γ≤2r−ℓ\operatorname{findim}\Gamma\le 2r-\ell, where ℓ\ell is the greatest positive integer with Ext⁡Λℓ−1(Se,Se)≠0\operatorname{Ext}^{\ell-1}_\Lambda(S_e,S_e)\ne 0 (finite because pd⁡ΛSe<∞\operatorname{pd}_\Lambda S_e<\infty). Their proof transports a minimal projective resolution of a Γ\Gamma-module through the tensor functor Λ(1−e)⊗Γ−\Lambda(1-e)\otimes_\Gamma- and controls where the resulting complex can have cohomology; the control rests on the fact that for primitive ee the Yoneda Ext-algebra Y(e)=⨁i≥0Ext⁡Λi(Se,Se)Y(e)=\bigoplus_{i\ge 0}\operatorname{Ext}^i_\Lambda(S_e,S_e), a graded algebra whose indecomposable projectives all share the same graded Loewy length (uniform graded Loewy length; the graded Loewy length of a graded module M=(Mi)i∈ZM=(M_i)_{i\in\mathbb{Z}} is the difference r−sr-s between the largest degree rr with Mr≠0M_r\ne 0 and the smallest degree ss with Ms≠0M_s\ne 0), forcing a cohomological cutoff at degree ℓ\ell. For a non-primitive idempotent, SeS_e is a direct sum of simples and Y(e)Y(e) need not have uniform graded Loewy length, which is exactly where the proof breaks down. The authors state explicitly that extending the theorem to idempotents that are not necessarily primitive would, via Auslander's construction, give a complete proof of the FDC; they also note (Remark 3.2) that the alternative route through a recollement of derived categories (a triple of six functors linking D(Λ)D(\Lambda) with D(eΛe)D(e\Lambda e) and D(Λ/ΛeΛ)D(\Lambda/\Lambda e\Lambda)) requires that Λe\Lambda e have finite projective dimension as a right eΛee\Lambda e-module, a condition that is not automatic. Subsequent work of Klapproth (2023) generalized Bravo–Paquette's minimal-resolution construction to arbitrary idempotents of semiperfect noetherian basic rings (a semiperfect ring is one over which every finitely generated module has a projective cover) and, under finite-global-dimension hypotheses on the intermediate idempotent subrings, settled an Ext-vanishing conjecture of Ingalls and Paquette — but obtained no finitistic-dimension bound for arbitrary idempotents.

Problem Statement

Let Λ\Lambda be an Artin algebra, ee an idempotent of Λ\Lambda that is not assumed primitive, Se=Λe/rad⁡ΛeS_e=\Lambda e/\operatorname{rad}\Lambda e the associated semisimple module, Y(e)=⨁i≥0Ext⁡Λi(Se,Se)Y(e)=\bigoplus_{i\ge 0}\operatorname{Ext}^i_\Lambda(S_e,S_e) its Yoneda Ext-algebra, and Γ=(1−e)Λ(1−e)\Gamma=(1-e)\Lambda(1-e) the corresponding corner algebra. Assume pd⁡ΛSe<∞\operatorname{pd}_\Lambda S_e<\infty and findim⁡Λ=r<∞\operatorname{findim}\Lambda=r<\infty. Prove that findim⁡Γ<∞\operatorname{findim}\Gamma<\infty, generalizing the primitive-case bound findim⁡Γ≤2r−ℓ\operatorname{findim}\Gamma\le 2r-\ell of Bravo–Paquette (Theorem 2.5) to idempotents ee that are not necessarily primitive, where for primitive ee, ℓ\ell is the greatest positive integer with Ext⁡Λℓ−1(Se,Se)≠0\operatorname{Ext}^{\ell-1}_\Lambda(S_e,S_e)\ne 0. Any proof valid for arbitrary idempotents under these hypotheses yields, via Auslander's construction (every Artin algebra Γ\Gamma is (1−e)Λ(1−e)(1-e)\Lambda(1-e) with gl.dim⁡Λ<∞\operatorname{gl.dim}\Lambda<\infty, hence r<∞r<\infty and pd⁡ΛSe<∞\operatorname{pd}_\Lambda S_e<\infty), a complete proof of the finitistic dimension conjecture.

References

  1. Diego Bravo and Charles Paquette, Idempotent reduction for the finitistic dimension conjecture, Proc. Amer. Math. Soc. 148 (2020), no. 5, 1891–1900, DOI 10.1090/proc/14945; arXiv:1902.00317, https://doi.org/10.1090/proc/14945
  2. Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488, DOI 10.1090/s0002-9947-1960-0157984-8, https://doi.org/10.1090/s0002-9947-1960-0157984-8
  3. Osamu Iyama, Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1011–1014, DOI 10.1090/s0002-9939-02-06616-9, https://doi.org/10.1090/s0002-9939-02-06616-9
  4. Edward L. Green, Chrysostomos Psaroudakis and Øyvind Solberg, Reduction techniques for the finitistic dimension, Trans. Amer. Math. Soc. 374 (2021), no. 10, 6839–6879, DOI 10.1090/tran/8409, https://doi.org/10.1090/tran/8409
  5. Carlo Klapproth, A method for constructing minimal projective resolutions over idempotent subrings, Proc. Amer. Math. Soc. 151 (2023), no. 11, 4579–4592, DOI 10.1090/proc/16470; arXiv:2111.03311, https://doi.org/10.1090/proc/16470
  6. K. R. Fuller and M. Saorín, On the finitistic dimension conjecture for artinian rings, Manuscripta Math. 74 (1992), no. 2, 117–132, DOI 10.1007/bf02567662, https://doi.org/10.1007/bf02567662
  7. Changchang Xi, On the finitistic dimension conjecture, III: related to the pair eAe ⊆ A, J. Algebra 319 (2008), no. 9, 3666–3688, DOI 10.1016/j.jalgebra.2008.01.021, https://doi.org/10.1016/j.jalgebra.2008.01.021
  8. John William MacQuarrie and Fernando dos Reis Naves, Quotient bifinite extensions and the finitistic dimension conjecture, arXiv:2306.02539 (2023), https://arxiv.org/abs/2306.02539
  9. Karin Erdmann, Odysseas Giatagantzidis, Chrysostomos Psaroudakis and Øyvind Solberg, Monomial arrow removal and the finitistic dimension conjecture, arXiv:2506.23747 (2025), https://arxiv.org/abs/2506.23747
  10. Odysseas Giatagantzidis, Arrow reductions for the finitistic dimension conjecture, arXiv:2507.12978 (2025), https://arxiv.org/abs/2507.12978