For a finite dimensional algebra Λ\Lambda, the (little) finitistic dimension findim⁡(Λ)\operatorname{findim}(\Lambda) is the supremum of the projective dimensions of the finitely generated Λ\Lambda-modules of finite projective dimension, and the big finitistic dimension FinDim⁡(Λ)\operatorname{FinDim}(\Lambda) drops the finite generation requirement. Bass's finitistic dimension conjecture (1960) asserts that findim⁡(Λ)<∞\operatorname{findim}(\Lambda)<\infty for every finite dimensional algebra Λ\Lambda; the analogous big finitistic dimension conjecture is stronger. The finitistic dimension is known to be left–right asymmetric as a value, but Cummings (2022/2024) proved that finiteness symmetry is equivalent to the conjecture itself: the little (resp. big) finitistic dimension conjecture holds for all finite dimensional algebras if and only if findim⁡(Λ)<∞⇒findim⁡(Λop)<∞\operatorname{findim}(\Lambda)<\infty\Rightarrow\operatorname{findim}(\Lambda^{\mathrm{op}})<\infty (resp. the FinDim⁡\operatorname{FinDim} analogue) for every finite dimensional algebra, and likewise the statement 'injectives generate the unbounded derived category of Λ\Lambda implies injectives generate for Λop\Lambda^{\mathrm{op}}' is equivalent to injectives generating for all finite dimensional algebras. The problem is to settle these implications — equivalently the underlying conjectures — in either direction; any counterexample would yield an algebra with infinite finitistic dimension on one side and vanishing finitistic dimension on the other. As of August 2026 the question remains open.

Background

A finite dimensional algebra is an associative unital algebra Λ\Lambda of finite dimension over a field kk; its modules are taken here as right modules, and Λop\Lambda^{\mathrm{op}} denotes the opposite algebra, whose right modules are the left Λ\Lambda-modules. The projective dimension pd⁡Λ(M)\operatorname{pd}_\Lambda(M) of a module MM is the minimal length of a projective resolution of MM (or ∞\infty if none is finite). Bass (1960) introduced two numerical invariants: the little finitistic dimension findim⁡(Λ)=sup⁡{pd⁡Λ(M):M finitely generated,pd⁡Λ(M)<∞}\operatorname{findim}(\Lambda)=\sup\{\operatorname{pd}_\Lambda(M) : M \text{ finitely generated}, \operatorname{pd}_\Lambda(M)<\infty\} and the big finitistic dimension FinDim⁡(Λ)\operatorname{FinDim}(\Lambda), the same supremum over all (possibly infinitely generated) right Λ\Lambda-modules of finite projective dimension. Since k[ℓ]/⟨ℓ2⟩k[\ell]/\langle\ell^2\rangle already has infinite global dimension — the global dimension is the supremum of the projective dimensions of all modules — this invariant is too coarse, and Bass conjectured that findim⁡(Λ)<∞\operatorname{findim}(\Lambda)<\infty for every finite dimensional (more generally, Artin) algebra — the finitistic dimension conjecture, one of the central homological conjectures of the representation theory of algebras, open since 1960; the stronger claim FinDim⁡(Λ)<∞\operatorname{FinDim}(\Lambda)<\infty for all such Λ\Lambda is the big finitistic dimension conjecture. The conjecture is known for commutative algebras; for monomial-relation algebras, meaning algebras presented by a quiver (a directed graph whose paths span the algebra) with defining relations that are single paths (Green–Kirkman–Kuzmanovich 1991); for algebras whose Jacobson radical JJ satisfies J3=0J^3=0; and for algebras of representation dimension at most three, the representation dimension being Auslander's measure of the homological complexity of the module category; but it remains open in general. A related 'first' conjecture, FinDim⁡=findim⁡\operatorname{FinDim}=\operatorname{findim}, is false. While the global dimension of a noetherian ring and the dominant dimension of a finite dimensional algebra are left–right symmetric invariants, the finitistic dimension is not: Jensen–Lenzing (1982) gave quiver algebras whose left and right finitistic dimensions differ, and Green–Kirkman–Kuzmanovich showed these two values can be arbitrarily far apart. This leaves open the weaker question of symmetry of finiteness, posed by Happel in 1991 and recorded in print by Christensen–Holm (2010, p. 27): if a finite dimensional algebra has finite finitistic dimension, must its opposite algebra also have finite finitistic dimension? Cummings answered this question 'heuristically': by adjoining to a basic algebra AA (every finite dimensional algebra being Morita equivalent, that is, having an equivalent module category, to a basic one, whose simple modules are one-dimensional over a splitting field), for each simple module, a copy of the dual-number algebra k[ℓ~]/⟨ℓ~2⟩k[\tilde\ell]/\langle\tilde\ell^2\rangle in a triangular matrix ring construction A~\tilde A, one has findim⁡(A)≤findim⁡(A~)\operatorname{findim}(A)\le\operatorname{findim}(\tilde A) and FinDim⁡(A)≤FinDim⁡(A~)\operatorname{FinDim}(A)\le\operatorname{FinDim}(\tilde A), while findim⁡(A~op)=FinDim⁡(A~op)=0\operatorname{findim}(\tilde A^{\mathrm{op}})=\operatorname{FinDim}(\tilde A^{\mathrm{op}})=0; hence Happel's finiteness-symmetry question is equivalent to the finitistic dimension conjecture itself (Theorem 3.4), and analogously for the big finitistic dimension (Theorem 3.5). The same construction governs a derived-category property: writing D(Λ)D(\Lambda) for the unbounded derived category of right Λ\Lambda-modules and Inj⁡-Λ\operatorname{Inj}\text{-}\Lambda for the injective modules, say that injectives generate for Λ\Lambda when the smallest localising (coproduct-closed triangulated) subcategory of D(Λ)D(\Lambda) containing Inj⁡-Λ\operatorname{Inj}\text{-}\Lambda is all of D(Λ)D(\Lambda). Rickard (2018/2019) proved that if injectives generate for a finite dimensional algebra then its finitistic dimension conjecture holds (indeed the big one), and asked whether injectives generate for every finite dimensional algebra — true for noetherian commutative rings, false for some non-noetherian commutative rings. Using that injectives generate for A~op\tilde A^{\mathrm{op}}, Cummings proved that 'injectives generate for all finite dimensional algebras' is equivalent to 'injectives generate for Λ\Lambda implies injectives generate for Λop\Lambda^{\mathrm{op}}, for every Λ\Lambda' (Theorem 4.6), with the dual statement for projective cogeneration also holding.

Problem Statement

Decide, for finite dimensional algebras, whether one-sided finiteness of the finitistic dimension forces finiteness on the other side — a question equivalent to the finitistic dimension conjecture itself. Specifically, where Λ\Lambda ranges over all finite dimensional algebras over arbitrary fields: (i) Is it true that findim⁡(Λ)<∞\operatorname{findim}(\Lambda)<\infty implies findim⁡(Λop)<∞\operatorname{findim}(\Lambda^{\mathrm{op}})<\infty? By Cummings's Theorem 3.4 this holds for all Λ\Lambda if and only if the little finitistic dimension conjecture holds, i.e. findim⁡(Λ)<∞\operatorname{findim}(\Lambda)<\infty for every finite dimensional algebra Λ\Lambda. (ii) The analogous question for the big finitistic dimension: does FinDim⁡(Λ)<∞\operatorname{FinDim}(\Lambda)<\infty imply FinDim⁡(Λop)<∞\operatorname{FinDim}(\Lambda^{\mathrm{op}})<\infty for all Λ\Lambda, equivalently, does FinDim⁡(Λ)<∞\operatorname{FinDim}(\Lambda)<\infty hold for every finite dimensional algebra? (iii) The derived-category analogue of Cummings's Theorem 4.6: if injectives generate the unbounded derived category D(Λ)D(\Lambda), must injectives generate D(Λop)D(\Lambda^{\mathrm{op}}), equivalently, do injectives generate for every finite dimensional algebra? These three questions share one objective — to determine whether the finiteness phenomena measured by the finitistic dimension and by injective generation are left–right symmetric in the class of finite dimensional algebras. They are not of equal strength: injective generation for all finite dimensional algebras (iii) implies the big finitistic dimension conjecture (ii), which implies the little one (i) — the first because injective generation forces the big finitistic dimension conjecture (Rickard) and the second because findim⁡≤FinDim⁡\operatorname{findim}\le\operatorname{FinDim} — so (iii) ⇒\Rightarrow (ii) ⇒\Rightarrow (i); dually, a counterexample to a weaker part refutes every stronger part.

References

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