8 problems

Let $A$ be a not necessarily commutative algebra of finite rank over a field $k$, and let $A$ be viewed as a left $A$-module. Nakayama's conjecture. If $A$ has an injective resolut…

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

Let $A$ be a finite dimensional algebra over an algebraically closed field. A brick is an $A$-module whose endomorphism ring is a division ring, and a generic module is an indecomp…

Let $A$ be an Artin algebra, let $J$ denote its Jacobson radical, and write $\operatorname{gldim}(A)$ for its global dimension and $\operatorname{id}(J)$ for the injective dimensio…

Let $S$ be a simple module over an artin algebra. Igusa–Liu–Paquette extension conjecture. If … then … for infinitely many integers $i$. This is presented as a stronger statement r…

Let $A$ be an artin algebra of infinite representation type. For an $A$-module $M$, its endolength is its length as a module over its endomorphism ring. Brauer–Thrall II. There exi…

Let $\Lambda$ be an artin algebra and let $S$ be a simple $\Lambda$-module. Strong no loop conjecture. If … then $\operatorname{Ext}^i_\Lambda(S,S)\neq 0$ for infinitely many integ…

Auslander–Reiten's generalized Nakayama conjecture. Any indecomposable injective $\Lambda$-module appears as a direct summand in the minimal injective resolution of $\Lambda$.