31 problems

Brauer–Thrall conjecture. The following statements hold:

Let $A$ and $B$ be finite-dimensional algebras over an algebraically closed field, and suppose they are stably equivalent, meaning their stable module categories are equivalent. Al…

Let $\mathsf{A}$ be a finite-dimensional, indecomposable, unital commutative operator algebra of minimal quotient complexity. Here, minimal quotient complexity means that every $r$…

Converse conjecture. If every semibrick is left finite, then $A$ is brick finite.

Let $B$ be a finite-dimensional algebra over an algebraically closed field $k$. Write $\operatorname{HH}_n(B)$ for the Hochschild homology groups of $B$. The algebra $B$ is (homolo…

Let $\Lambda$ be a finite-dimensional algebra. A semibrick in $\operatorname{mod} \Lambda$ is a set of bricks whose distinct members have no nonzero morphisms between them. Enomoto…

Silting mutation conjecture. Then $T$ is tilting and $\operatorname{End}_A(T)$ is a $d$-representation infinite algebra.

Gorenstein bimodule conjecture. $A$ is self-injective if and only if $A$, regarded as an $A$-bimodule, is Gorenstein-projective.

Let $A$ be a finite-dimensional algebra over a field. An Auslander-Reiten sequence is called $n$-torsion-free when it has the torsion-freeness property of order $n$ considered in t…

Let $A$ be a finite-dimensional algebra over a field. The condition that all higher extensions from its $k$-dual back to $A$ vanish is … Here $D(A)$ denotes the $k$-dual of $A$. Ta…

Uniqueness conjecture. The statement of Theorem $$ holds for all finite-dimensional algebras.

Tau-tilting finiteness–brick-discreteness conjecture. The following are equivalent:

Brick-infinite equivalence conjecture. The following are equivalent:

Let $A$ be a finite-dimensional algebra. Say that $A$ has discrete general representation type when every irreducible component of each representation variety for $A$ has a dense o…

Generalized Kaplansky conjecture. The evaluation of any multilinear polynomial over $A$ is a vector subspace of $A$.

Let $A$ be an algebra, and let $\tau$-tilting theory be considered over a field $\mathbb{K}$. An algebra is of finite representation type when it has only finitely many isomorphism…

Semibrick finiteness conjecture. If every semibrick in $\operatorname{mod}\Lambda$ is a finite set, then $\operatorname{mod}\Lambda$ is brick-finite, that is, $\Lambda$ is $\tau$-t…

DO and Schur-representation finiteness conjectures. The following hold:

Vanishing upper Serre dimension conjecture. The upper Serre dimension satisfies

Simson's Brauer-Thrall 3 conjecture. For any infinite cardinality $\lambda$, there is an indecomposable $A$-module $M$ of dimension

Bongartz–Ringel accessibility conjecture. Every indecomposable module of finite length over any algebra is accessible.

Finiteness conjecture. For a given finite-dimensional $k$-algebra $A$, there are only finitely many integers $d$ such that $A$ is $d$-representation-finite.

Let $k$ be a field and let $A$ be a finite-dimensional $k$-algebra. The module category $A\text{-}\operatorname{mod}$ is D-standard over $k$ if every pseudo-identity on…

Semisimple-quotient conjecture. For almost all $v\in\mathcal V_k$,

Dominant-dimension property conjecture. Every finite dimensional algebra with finite dominant dimension $n\geq 1$ has property $*$.