Injective generation for finite dimensional algebras: do injective modules generate the unbounded derived category?

For a finite dimensional algebra $A$ over a field, let $\mathcal{D}(A)$ be the unbounded derived category of all right $A$-modules. Rickard (2019) highlighted the question, first circulated by Keller in 2001, whether the injective $A$-modules generate $\mathcal{D}(A)$ as a localizing subcategory. He proved that injectives generate for every commutative Noetherian ring but not for a suitable commutative non-Noetherian ring, and that for a finite dimensional algebra injective generation forces the big finitistic dimension to be finite, so a positive answer for all finite dimensional algebras would imply the finitistic dimension conjecture. No finite dimensional algebra is known for which injectives fail to generate. No counterexample and no universal proof are known: the question remains open, with 2024–2026 work still producing new positive classes and equivalent reformulations.

Background

A finite dimensional algebra $A$ is an associative algebra over a field $k$ that has finite dimension as a $k$-vector space; one studies its module categories and homological invariants. The unbounded derived category $\mathcal{D}(A)$ is the category obtained from cochain complexes of right $A$-modules, with no boundedness restriction on the degrees, by formally inverting quasi-isomorphisms (maps inducing isomorphisms on cohomology); it is a triangulated category (an additive category with a shift autoequivalence $X\mapsto X[1]$ and a class of distinguished triangles satisfying the standard axioms, so that mapping-cone and long-exact-sequence arguments are available) with arbitrary set-indexed coproducts. A localizing subcategory is a full triangulated subcategory closed under such coproducts, and every class of objects generates a smallest one; the statement that injectives generate for $A$ means that the smallest localizing subcategory containing all injective right $A$-modules (modules $E$ for which the functor $\operatorname{Hom}_A(-,E)$ is exact) is the whole of $\mathcal{D}(A)$. By classical results, the projective modules always generate $\mathcal{D}(A)$ for any ring, but the dual statement for injectives can fail. The finitistic dimension conjecture, going back to Bass (1960), asserts that the little finitistic dimension $\operatorname{findim}(A)$ — the supremum of the projective dimensions of the finitely generated $A$-modules of finite projective dimension — is finite for every finite dimensional algebra; the big finitistic dimension $\operatorname{FPD}(A)$ is the same supremum over all modules and its finiteness for every finite dimensional algebra is an even stronger open statement. Rickard's 2019 paper proved: injectives generate for every commutative Noetherian ring (Theorem 3.3); there is a commutative non-Noetherian ring for which they do not (Theorem 3.5); for finite dimensional algebras, injectives generate whenever $A$ is commutative, Gorenstein (i.e. projectives have finite injective dimension and injectives have finite projective dimension), of finite global dimension (every module has finite projective dimension), self-injective (every projective module is injective), of finite representation type (only finitely many indecomposable finitely generated modules, up to isomorphism), of radical square zero (the Jacobson radical $J$ of $A$ — the intersection of its maximal right ideals — satisfies $J^2=0$), monomial (a quotient of a path algebra — the algebra spanned by the paths of a directed graph — by an ideal generated by paths), or derived equivalent (having a derived category equivalent as a triangulated category) to an algebra in one of these classes (Theorem 8.1); and if injectives generate for a finite dimensional algebra $A$, then $\operatorname{FPD}(A)$ is finite (Theorem 4.3). The paper's final section records that no finite dimensional algebra is known for which injectives fail to generate, and that injective generation is likewise open for local algebras (those with a unique maximal right ideal, equivalently $A/J\cong k$), for algebras with radical cube zero ($J^3=0$, a class where the little finitistic dimension conjecture is known), and for algebras of representation dimension at most three (the representation dimension, introduced by Auslander, is the infimum of the global dimensions of endomorphism rings of modules that are both generators and cogenerators; this is another class where the little conjecture is known). The injective generation question for finite dimensional algebras is therefore a self-contained strengthening of the finitistic dimension conjecture: it asks for a structural, derived-categorical reason for finiteness, and for algebras satisfying the mild additional condition of Eshraghi-Hajizamani (2024), under which injective generation and the big finitistic dimension conjecture are equivalent, its failure anywhere would refute the big finitistic dimension conjecture for that algebra.

Problem Statement

Decide whether injective modules generate the unbounded derived category of every finite dimensional algebra. Formally: for a finite dimensional algebra $A$ over a field, let $\mathcal{D}(A)$ be the unbounded derived category of right $A$-modules and $\operatorname{Loc}(\operatorname{Inj-}A)$ the smallest localizing subcategory of $\mathcal{D}(A)$ containing all injective $A$-modules. Is $\operatorname{Loc}(\operatorname{Inj-}A)=\mathcal{D}(A)$ for every such $A$? Refuting injective generation for a finite dimensional algebra $A$ amounts to exhibiting a nonzero object $X$ of $\mathcal{D}(A)$ such that $\operatorname{Hom}_{\mathcal{D}(A)}(E,X[n])=0$ for every injective $A$-module $E$ and every integer $n$, and by Theorem 4.3 recorded above it also follows for any finite dimensional algebra with infinite big finitistic dimension; affirming it amounts to a proof that injectives generate for every finite dimensional algebra over every field. A positive answer would in particular settle the question for the classes the source leaves open (local algebras, algebras with radical cube zero, algebras of representation dimension at most three) and would imply the big, hence the little, finitistic dimension conjecture for all finite dimensional algebras; a negative answer would give the first counterexample to injective generation in the finite dimensional world, and would refute the big finitistic dimension conjecture for that algebra whenever it satisfies the mild additional condition of Eshraghi-Hajizamani (2024), under which the two properties are equivalent. The source's adjacent open questions (a right Noetherian noncommutative ring without injective generation, the precise relation between localizing generation by injectives and colocalizing generation by projectives, and whether the two halves of the Gorenstein condition imply each other) are separate problems and do not belong to this problem's completion boundary.

References

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