Existence of radical-preserving homological maps from basic Artin algebras to algebras of finite finitistic dimension (an equivalent form of the Finitistic Dimension Conjecture)

For every basic Artin algebra $A$ , does there exist a basic Artin algebra $B$ of finite little finitistic dimension and a ring homomorphism $\phi:A\to B$ whose kernel is superfluous (contained in the Jacobson radical of $A$ ) and such that $B$ , viewed as a right $A$ -module via $\phi$ , has finite flat dimension? Giatagantzidis (2024, published 2025 in the Bulletin of the London Mathematical Society) proved that a positive answer for all basic Artin algebras is equivalent to the Finitistic Dimension Conjecture, and conversely that a single basic Artin algebra admitting no such pair $(B,\phi)$ would refute the conjecture. The paper constructs such maps only for algebras of quasi-uniform Loewy length via an explicit uniformization monomorphism; the general existence problem, being equivalent to one of the oldest open problems in the homological algebra of Artin algebras, remains open.

Background

An Artin algebra is an algebra that is finitely generated as a module over a commutative artinian ring; the central case is a finite-dimensional algebra over a field $k$ , which by standard structure theory can be presented as a bound quiver algebra $kQ/I$ : the path algebra of a finite directed graph $Q$ (the quiver, whose basis elements are the paths in $Q$ ) modulo an admissible ideal $I$ of relations (an ideal containing all sufficiently long paths and consisting only of combinations of paths of length at least two, which makes $kQ/I$ finite-dimensional). A basic Artin algebra is one whose indecomposable projective modules are pairwise non-isomorphic; every Artin algebra is Morita equivalent (equivalent for representation-theoretic purposes) to a unique basic one, so restricting to basic algebras loses no generality.

The homological complexity of an algebra $A$ is measured by homological dimensions. The global dimension is the supremum of the projective dimensions of all $A$ -modules, where the projective dimension of a module is the length of its shortest resolution by projective modules; when the global dimension is infinite, softer invariants are used. The little finitistic dimension $\operatorname{fin.dim}A$ is the supremum of the projective dimensions of finitely generated left $A$ -modules of finite projective dimension, and the big finitistic dimension $\operatorname{Fin.dim}A$ takes the same supremum over all left modules. The Finitistic Dimension Conjecture (FDC), publicized by H. Bass in 1960, predicts that $\operatorname{fin.dim}A$ is finite for every Artin algebra $A$ . It has been confirmed for many special classes (monomial algebras, algebras whose radical cube vanishes, and others) and has resisted general attack for over six decades; no counterexample is known.

A persistent strategy is change of rings: bound the finitistic dimension of $A$ through a ring homomorphism $\phi:A\to B$ . Small's 1968 change-of-rings theorem covers surjective $\phi:A\twoheadrightarrow A/K$ with superfluous kernel $K$ , that is $K\subseteq J(A)$ where $J(A)$ denotes the Jacobson radical (the intersection of all maximal left ideals), giving $\operatorname{fin.dim}A\le\operatorname{fin.dim}B+\operatorname{pd}B_A$ with $B_A$ the right $A$ -module obtained by restriction of scalars. Giatagantzidis (2024/2025) extended this to non-surjective maps: a homomorphism $\phi:A\to B$ is radical-preserving if $\phi(J(A))\subseteq J(B)$ , its kernel is superfluous if it lies in $J(A)$ , and the key inequality $\operatorname{fin.dim}A\le\operatorname{fin.dim}B+\operatorname{fd}B_A$ (with analogues for the big finitistic and global dimensions) holds for radical-preserving homomorphisms of left artinian rings with superfluous kernel and finite flat dimension $\operatorname{fd}B_A$ . Here the flat dimension is the length of the shortest resolution of $B_A$ by flat modules (modules over which the tensor product functor is exact), an invariant that agrees with the projective dimension for finitely generated modules over an Artin algebra. Two facts make this a reduction of the FDC. First, radical preservation is automatic in many situations: every homomorphism of Artin algebras whose target $B$ is basic is radical-preserving, since the Jacobson radical of a basic semiperfect ring (a ring over which every finitely generated module has a projective cover) contains all nilpotent elements. Second, the paper constructs, for every bound quiver algebra $\Lambda$ , a uniformization algebra $\Lambda^{*}$ with $\operatorname{Fin.dim}\Lambda^{*}=0$ and a radical-preserving monomorphism $\iota:\Lambda\hookrightarrow\Lambda^{*}$ , obtained by adding one loop at each vertex whose indecomposable projective module has non-maximal Loewy length (the Loewy length of a module is the number of steps in its radical filtration $M\supseteq JM\supseteq J^{2}M\supseteq\cdots$ ). The monomorphism $\iota$ has finite projective dimension precisely when $\Lambda$ has quasi-uniform Loewy length, meaning every indecomposable projective $\Lambda$ -module whose top has infinite injective dimension has maximal Loewy length; this yields the paper's theorem that all such algebras satisfy the FDC. But for general $\Lambda$ the construction gives no finite-dimension guarantee, and the author stresses that establishing such homomorphisms in general is, in his words, not an easy task. Combining the inequality with the automatic radical preservation gives the paper's equivalent reformulation of the conjecture: the FDC holds for Artin algebras if and only if for every basic Artin algebra $A$ there exist a basic Artin algebra $B$ with $\operatorname{fin.dim}B<\infty$ and a homomorphism $\phi:A\to B$ with superfluous kernel and finite projective dimension (Corollary 3.15 of the published version; numbered Corollary 3.14 in the first arXiv version); since $B_A$ is finitely generated, flat and projective dimension coincide there, so this is the same condition as the finite flat dimension $\operatorname{fd}B_A<\infty$ used in the problem statement below. The reformulation remains valid with Artin algebras replaced by finite-dimensional algebras over a field.

Problem Statement

For every basic Artin algebra $A$ , does there exist a basic Artin algebra $B$ with $\operatorname{fin.dim}B<\infty$ and a ring homomorphism $\phi:A\to B$ whose kernel is superfluous (that is, $\operatorname{Ker}\phi\subseteq J(A)$ ) and which has finite flat dimension, i.e. $\operatorname{fd}B_A<\infty$ for $B$ viewed as a right $A$ -module by restriction of scalars along $\phi$ ? Since every homomorphism of Artin algebras with basic target is automatically radical-preserving ( $\phi(J(A))\subseteq J(B)$ ), this is precisely the existence problem to which Giatagantzidis reduced the Finitistic Dimension Conjecture: by Corollary 3.15 of the published version (Corollary 3.14 of arXiv v1), the FDC holds for Artin algebras if and only if the answer is yes for every basic Artin algebra, and the equivalence persists verbatim for finite-dimensional algebras over a field. Establishing such a pair $(B,\phi)$ for every basic Artin algebra would prove the FDC, while exhibiting a basic Artin algebra $A$ that admits no such pair would yield the first counterexample to the FDC. The problem is an equivalent reformulation of the famous conjecture, not a restricted variant of it: solving it in either direction resolves the FDC for Artin algebras exactly at its original generality.

References

Odysseas Giatagantzidis, Radical preservation and the finitistic dimension, Bulletin of the London Mathematical Society 58 (1), published online 3 November 2025, DOI 10.1112/blms.70222, https://doi.org/10.1112/blms.70222 (preprint arXiv:2412.10965, https://arxiv.org/abs/2412.10965)
Odysseas Giatagantzidis, Arrow reductions for the finitistic dimension conjecture, arXiv:2507.12978 (2025), https://arxiv.org/abs/2507.12978
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
Edward Green, Chrysostomos Psaroudakis, Øyvind Solberg, Reduction techniques for the finitistic dimension, Transactions of the American Mathematical Society 374 (2021), 6839–6879, DOI 10.1090/tran/8409, https://doi.org/10.1090/tran/8409
John MacQuarrie, Fernando dos Reis Naves, Quotient bifinite extensions and the finitistic dimension conjecture, Proceedings of the American Mathematical Society 152 (2024), 585–590, DOI 10.1090/proc/16569, https://doi.org/10.1090/proc/16569
L.W. Small, A change of rings theorem, Proceedings of the American Mathematical Society 19 (1968), 662–666, DOI 10.1090/s0002-9939-1968-0223412-x, https://doi.org/10.1090/s0002-9939-1968-0223412-x
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
Panagiotis Kostas, Finitistic dimension via modules over the singularity category, arXiv:2608.15482 (2026), https://arxiv.org/abs/2608.15482