14 problems

Let $A$ be a connected Artin–Schelter regular algebra, meaning a connected graded regular algebra of polynomial growth. Artin–Schelter's noetherianity conjecture. Every connected A…

Let $A$ be an Artin-Schelter regular algebra, and let $\nu_A$ denote its Nakayama automorphism. The homological determinant $\operatorname{hdet}(\nu_A)$ is the homological determin…

Hilbert-series conjecture. Theorems A and B are true for all three-dimensional Artin–Schelter regular algebras generated in degree one, not only for the generic ones.

Let $A$ be an Artin–Schelter regular algebra, or a noetherian algebra of finite global dimension. Its Hilbert series is said to be that of a weighted polynomial ring when … for pos…

Let $A$ be an Artin–Schelter regular algebra of global dimension $d$. Consider the properties that $A$ is a noetherian domain, that its Hilbert series has the form … for positive i…

Let $A$ be the most generic quadratic Artin–Schelter regular algebra of dimension four with four generators and six relations, and let its line scheme be the parameter space of lin…

Let $A$ be an Artin–Schelter regular algebra of finite GK-dimension. Artin–Tate–Van den Bergh domain conjecture. Every such algebra is a domain. The conjecture is motivated by resu…

Let $A$ be a connected graded noetherian algebra of finite global dimension. Regularity conjecture. The algebra $A$ must be AS-regular. This conjecture is suggested by results in g…

Let $A$ be an AS-regular algebra of global dimension $d$. Consider the following properties: $A$ is noetherian; $A$ is a domain; $\operatorname{GKdim}(A)=d$; the minimal number of…

Let $X$ be a Calabi–Yau curve and let $ξ$ be a simple vector bundle on $X$. Let $N(ξ)$ be the coarse moduli scheme of the moduli stack of non-splitting complexes $O_X\to V$ such th…

Noncommutative Auslander conjecture. The map $\varphi$ is an isomorphism of graded algebras.

Let $A$ be an Artin–Schelter regular algebra and let a finite group or semisimple Hopf algebra act on $A$. The action has trivial homological determinant when its homological deter…

Line-scheme conjecture. The line scheme of the most generic quadratic regular algebra of global dimension four should be isomorphic to the union of two spatial elliptic curves and…

Let $A$ be a noncommutative Artin–Schelter regular algebra and let $G$ be a finite group acting on $A$ by graded algebra automorphisms. Write $A^G$ for the fixed subring. Shephard–…