66 problems

Nakayama's conjecture. If an Artin algebra has infinite dominant dimension, then it is self-injective.

Let $A$ be an Artin algebra. Its left and right self-injective dimensions are the injective dimensions of the regular module on the left and right, respectively. Gorenstein symmetr…

Let $A$ be an Artin algebra. The Krull–Gabriel dimension of $A$ is an invariant measuring the complexity of its module category, and $A$ is domestic in the usual representation-the…

Bass's finitistic dimension conjecture. For every artin algebra $R$,

Let $\Lambda$ be an Artin algebra, and write $\operatorname{id}_\Lambda\Lambda$ for its left self-injective dimension and $\operatorname{id}\Lambda_\Lambda$ for its right self-inje…

Let $R$ be an Artin algebra. The Finitistic Dimension Conjecture. The supremum of the projective dimension of all finitely generated left $R$-modules with finite projective dimensi…

Let $R$ be an Artin algebra, and let $R$ have a minimal injective resolution. An indecomposable injective left $R$-module is an injective left $R$-module that cannot be written as…

The finitistic dimension conjecture. For every Artin algebra $\Lambda$, both $\operatorname{findim}\Lambda$ and $\operatorname{Findim}\Lambda$ are finite.

Finitistic dimension conjecture. The little finitistic dimension $\operatorname{findim}(R)$ is finite for every Artin algebra $R$.

Let $R$ and $S$ be Artin algebras, let $_RU$ be a generalized tilting module, and set $S={\rm End}(_RU)$. A module is (quasi) Gorenstein when it satisfies the corresponding Gorenst…

Gorenstein Symmetric Conjecture. The left and right self-injective dimensions are identical for every artin algebra $\Lambda$:

Nakayama Conjecture. The conjectural assertion is not included in the supplied statement span.

Let $\Lambda$ be an artin algebra. For a finitely generated right $\Lambda$-module $N$, write…

Finitistic dimension conjecture. Every artin algebra $\Lambda$ satisfies

Auslander–Reiten–Smalø conjecture. If $A$ is of infinite representation type, then $\Gamma_A$ has infinitely many connected components.

Finite-length realization conjecture. We conjecture that

Finitistic dimension conjecture. Every Artin algebra has finite finitistic dimension.

Let $A$ be an Artin algebra, and let $\mathsf{fin.dim}A$ denote the supremum of the projective dimensions of all finitely generated right $A$-modules having finite projective dimen…

Compactly Gorenstein algebra conjecture. All Artin algebras should be compactly Gorenstein.

Let $\Lambda$ be an Artin algebra. The Auslander Gorenstein conjecture. If $\Lambda$ satisfies the Auslander condition, meaning that it is $n$-Gorenstein for every positive integer…

Finitistic dimension conjecture. The big finitistic dimension of every Artin algebra $\Lambda$ is finite.

Self-orthogonal Wakamatsu-tilting conjecture. If $|T|=|A|$, then $T$ is Wakamatsu-tilting.

Let $k$ be a commutative artinian ring, $A$ an Artin $k$-algebra, and let $\operatorname{mod} A$ denote the category of finite length left $A$-modules. Assume that…

Let $\Lambda$ be a non-semisimple artin algebra. The finitistic dimension of $\Lambda$ is the supremum of the projective dimensions of all finitely generated $\Lambda$-modules havi…

Schröer's ordinal conjecture.