The Finitistic Dimension Conjecture for Artin Algebras and its Equivalent Extension-Transfer Reformulations

The finitistic dimension conjecture (FDC), a question of Rosenberg and Zelinsky publicized by Bass in 1960, predicts that every Artin algebra — in particular every finite-dimensional algebra over a field — has finite finitistic dimension, i.e. only finitely many finite projective dimensions occur among its finitely generated modules. The conjecture has been open for more than six decades and implies the Nakayama, generalized Nakayama, strong Nakayama, Wakamatsu tilting, and Gorenstein symmetry conjectures. Xi and Xu (2013) recast FDC as a transfer statement for extensions of algebras: FDC is equivalent to the statement that whenever $B\subseteq A$ is an inclusion of finite-dimensional algebras whose smaller algebra's Jacobson radical is a left ideal of $A$ , finiteness of the finitistic dimension of $A$ forces finiteness of that of $B$ ; over a perfect field the conjecture is further reduced to extensions of relative global dimension at most one. The conjecture and all equivalent reformulations remain open through 2026; only sufficient conditions for finiteness transfer are known.

Background

An Artin algebra is a ring that is finitely generated as a module over its center, the latter being a commutative Artinian ring; the leading examples are finite-dimensional algebras over a field, which by Morita theory and the Gabriel theorem can be presented as bound quiver algebras $kQ/I$ (a path algebra of a finite quiver modulo an admissible ideal). For such an algebra $A$ , the (little) finitistic dimension $\mathrm{fin.dim}(A)$ is the supremum of the projective dimensions $\mathrm{pd}(M)$ of all finitely generated left $A$ -modules $M$ that have finite projective dimension, where the projective dimension is the minimal length of a projective resolution of $M$ . Projective dimensions of individual modules can be arbitrarily large, and the finitistic dimension asks whether the finite ones are globally bounded within the module category. The finitistic dimension conjecture (FDC) asserts that $\mathrm{fin.dim}(A)<\infty$ for every Artin algebra. It originated as a 1960 question of Rosenberg and Zelinsky publicized by Bass and has become one of the main open problems in the representation theory of algebras; it is known only for special classes of algebras (for example algebras with representation-finite or finite-global-dimension companions in the sense of Xi's FDC I and II papers), and its validity would imply the Nakayama conjecture, the generalized Nakayama conjecture, the strong Nakayama conjecture, the Wakamatsu tilting conjecture, and the Gorenstein symmetry conjecture. A fruitful attack, pioneered by Xi and continued with Xu, compares the finitistic dimensions of two algebras in an extension $B\subseteq A$ . For such an extension one defines a left $A$ -module $X$ to be $(A,B)$ -projective (relatively projective) if $X$ is a direct summand of $A\otimes_B X$ ; the relative global dimension $\mathrm{gl.dim}(A,B)$ is the supremum over all finitely generated $A$ -modules of the lengths of shortest resolutions by $(A,B)$ -projective modules. A syzygy of a module is the kernel at a given step of a projective (or relative) resolution. Xi and Xu proved: if $B\subseteq A$ is an inclusion of Artin algebras whose Jacobson radical $\mathrm{rad}(B)$ is a left ideal of $A$ and the category of $(A,B)$ -projective $A$ -modules is closed under taking $A$ -syzygies, then $\mathrm{fin.dim}(B)\le\mathrm{fin.dim}(A)+\mathrm{fin.dim}({}_B A)+3$ ; that $n$ -hereditary extensions satisfy $\mathrm{gl.dim}(A)\le\mathrm{gl.dim}(B)+n$ ; and that trivially twisted extensions of two algebras of finite finitistic dimension again have finite finitistic dimension. Crucially, their paper restates and proves that FDC is equivalent to a finiteness-transfer principle for extensions, giving a new formulation in terms of relative homological dimension: over any field $k$ , FDC is equivalent to the statement that for all inclusions $B\subseteq A$ of finite-dimensional $k$ -algebras with $\mathrm{rad}(B)$ a left ideal of $A$ , finiteness of $\mathrm{fin.dim}(A)$ implies finiteness of $\mathrm{fin.dim}(B)$ (an equivalence originating in Xi's FDC I and II papers); and, by a different method working over a perfect field, FDC is reduced to the corresponding transfer statement for extensions of relative global dimension at most one, i.e. with $\mathrm{gl.dim}(A,B)\le 1$ . Here a perfect field is one over which every algebraic extension is separable (e.g. any field of characteristic zero or any finite field). Subsequent work of Wang–Xi on radical-power extensions, Guo on relative finitistic dimensions, MacQuarrie–Naves on quotient bifinite extensions, and Giatagantzidis on radical-preserving homomorphisms has enlarged the stock of sufficient conditions, but none of the equivalent forms of the conjecture is settled.

Problem Statement

Is the finitistic dimension of every Artin algebra finite? Equivalently (Xi–Xu 2013, building on Xi's FDC I and II): is it the case that for every field $k$ and every inclusion $B\subseteq A$ of finite-dimensional $k$ -algebras such that the Jacobson radical $\mathrm{rad}(B)$ is a left ideal of $A$ , if $\mathrm{fin.dim}(A)<\infty$ then $\mathrm{fin.dim}(B)<\infty$ ? By the same paper's relative-homological reformulation, over a perfect field this is equivalent (in reduction form) to the corresponding finiteness-transfer statement for extensions of relative global dimension $\mathrm{gl.dim}(A,B)\le 1$ . The conjecture would be refuted by an Artin algebra — concretely, a finite-dimensional algebra over a field — with finitely generated modules of arbitrarily large finite projective dimension; by the equivalences above, such an algebra refutes both transfer statements as well.

References

Changchang Xi and Dengming Xu, 'The finitistic dimension conjecture and relatively projective modules', Communications in Contemporary Mathematics 15 (2013), no. 2, 1350004. DOI: 10.1142/S0219199713500041, https://doi.org/10.1142/S0219199713500041
Hyman Bass, 'Finitistic dimension and a homological generalization of semi-primary rings', Transactions of the American Mathematical Society 95 (1960), no. 3, 466–488. DOI: 10.1090/s0002-9947-1960-0157984-8, https://doi.org/10.1090/s0002-9947-1960-0157984-8
Changchang Xi, 'On the finitistic dimension conjecture I: related to representation-finite algebras', Journal of Pure and Applied Algebra 193 (2004), nos. 1–3, 287–305. DOI: 10.1016/j.jpaa.2004.03.009, https://doi.org/10.1016/j.jpaa.2004.03.009
Changchang Xi, 'On the finitistic dimension conjecture II: related to finite global dimension', Advances in Mathematics 201 (2006), no. 1, 116–142. DOI: 10.1016/j.aim.2005.02.002, https://doi.org/10.1016/j.aim.2005.02.002
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John William MacQuarrie and Fernando dos Reis Naves, 'Quotient bifinite extensions and the finitistic dimension conjecture', Proceedings of the American Mathematical Society 152 (2024), no. 2. DOI: 10.1090/proc/16569, arXiv:2306.02539, https://arxiv.org/abs/2306.02539
Odysseas Giatagantzidis, 'Radical preservation and the finitistic dimension', Bulletin of the London Mathematical Society 58 (2026), no. 1, article e70222 (published online 3 November 2025). DOI: 10.1112/blms.70222, arXiv:2412.10965, https://arxiv.org/abs/2412.10965
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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
Odysseas Giatagantzidis, 'Arrow reductions for the finitistic dimension conjecture', arXiv:2507.12978 (2025), https://arxiv.org/abs/2507.12978