15 problems

Non-kissing extremal-pair conjecture. 1. The $\boldsymbol{c}$-vector $\boldsymbol{m}_\sigma$ is the direction of at least one extremal exchangeable pair. 2. If $\omega$ and…

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Say that $A$ has a well-defined Auslander–Reiten map if every indecomposable injective $A$-modul…

Torsion-free Auslander–Reiten sequence conjecture. $A$ has $n$-torsion-free Auslander–Reiten sequences for every $n\in\mathbb{N}$ if and only if $A$ is self-injective.

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…

Auslander–Reiten quiver component conjecture. If $\Gamma$ is of infinite representation type, then its AR-quiver has infinitely many components.

Let $A$ be a Brauer tree algebra, let $u$ be its associated graded algebra, and let $overline{nu}$ be the algebra considered in the source. Write $Gamma$ for the Auslander–Reiten…

Let $(Q,\mathcal{S})$ be a Jacobi-finite quiver with potential, and let a decorated representation mean a representation together with its decorated part. A decorated representatio…

First-level Auslander–Reiten conjecture. The module $\tau M$ is also of the first level.

Let $\mathbb X$ be a non-singular proper scheme of dimension $d\geq 1$ over a field $k$, let $C$ be a coherent $\mathcal O_{\mathbb X}$-module, and let $\tau(C,k)$ denote the Ausla…

Let $\Lambda$ be an artin algebra with center $k$, let $D$ be the duality functor associated with a minimal cogenerator of $\operatorname{mod} k$, and let $K$ and $Y$ be modules. W…

The authors' conjecture. The connected components of \text{mathitGamma}_{{\rm rep}(Q)} containing representations not in $\overline{\mystrut(5,0) {\rm rep\hspace{-1pt}}}\,(Q)$…

Auslander–Reiten component conjecture. If $A$ is of infinite representation type, then its Auslander–Reiten quiver has infinitely many Auslander–Reiten components.

Let $A$ be a simply connected Gorenstein differential graded algebra of finite type. Write $\operatorname{H}^eA$ and $\operatorname{H}^*A$ for its cohomology in degree $e$ and in a…

Let $\Lambda$ be a triangular algebra and let $U$ be a uniserial $\Lambda$-module with mast $p$. Write $J$ for the Jacobson radical, and let $B$, $C$, and $D$ denote the sets of ar…

Conjecture. For any simple module $S_x$, there is a periodic module $W$ with