56 problems

Bahturin–Regev conjecture. The following statements hold:

Let $A$ be a regular graded algebra, meaning that it has finite global dimension and satisfies the Gorenstein condition … The algebra $A$ is graded coherent if every finitely gener…

Let $A$ be a connected finitely generated ${\mathbb N}$-graded algebra over a field $k$, with $A_0=k$. The algebra $A$ is graded coherent if every finitely generated graded ideal i…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, let $I\subset R$ be a graded ideal, and set $c=\operatorname{codim} R/I$. For a minimal graded free reso…

The $e_m$-overkiller conjecture. For $3\leq m\leq k$,

The exact $e_m$-killer conjecture. For $1\leq m\leq k$,

The filtration-saturation conjecture. For $1\leq m\leq k$,

Let $C$ be a hyperkähler cone, and let $B$ be the graded H-algebra defined in Definition 3.3. SGH-algebra conjecture. The graded H-algebra $B$ is an SGH-algebra. This is the algebr…

Realization conjecture. There exist an exact category $\mathcal{B}$ of finite homological dimension and a group $H$ acting on $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ such that…

Nonexistence conjecture. No nontrivial $G$-grading $\Gamma$ on $M_n(K)$ satisfies the primeness property for graded central polynomials.

Let $E$ be an imaginary quadratic field with odd discriminant $-D$, let $(L,\langle\phantom{x},\phantom{x}\rangle)$ be a Hermitian lattice over $\mathcal{O}_E$ of signature $(1,n)$…

Consider a reduced Gorenstein standard graded algebra with Krull dimension at least its socle dimension, and consider a generic Artinian reduction after passing to an appropriate f…

Let $A$ be a domain equipped with an injective well-ordered valuation $\nu$, and let $gr A$ denote the associated graded algebra of the filtration on $A$ induced by $\nu$. A family…

Let $K$ be a field and let $n$ be a prime number. Let $K\langle X\mid \mathbb{Z}_n\rangle$ denote the free $\mathbb{Z}_n$-graded algebra, and let…

Let $S_n$ be the symmetric group, let $Z(\mathbb{C}S_n)$ be its center, and let $\Theta\colon Z(\mathbb{C}S_n)\xrightarrow{\sim}\mathbb{C}[\mathcal{C}(n)]^{\mathbb{T}}$ be the isom…

Let $G$ be a pseudo-reflection group. Write $\mathcal{SH}_G^{i,k}$ for the component of $x$-degree $i$ and $\theta$-degree $k$, let $\Delta_G$ be the discriminant, and let…

Let $C_{u,1-u}$ be the zinbiel sub-algebra parametrized by $u$, and let $\Pi:C_{u,1-u}\to\mathsf{A}_{\mathsf{MZV}}$ be the corresponding morphism of commutative algebras. Generic i…

Halperin's conjecture. If $H^*$ is a positively elliptic algebra, then $H^*$ does not admit a non-trivial derivation of negative degree.

Let $\mathcal{L}(K)$ be the $K$-algebra generated by the lambda values, equipped with the lambda-degree induced by the lambda-type of composition arrays and taking values in the mo…

Let $A$ be a noetherian connected graded algebra with a balanced dualizing complex. Hypothesis … . Rationality over $\mathbb{Q}$ is known for many classes of graded algebras, inclu…

Let $\operatorname{\mathbbm{k}}$ be an algebraically closed field, and let $R$ be a graded domain of Krull dimension one over $\operatorname{\mathbbm{k}}$. Call $R$ cyclotomic when…

Generalized Rankin–Cohen proposition. The proposition concerning Rankin–Cohen algebras holds for such a graded algebra; it is not necessary that $M_0={\sf k}.1$. The paper identifi…

Generalized-action PI-exponent conjecture. If $L$ is $H$-nice, then there exist constants $C\in\mathbb{R}$, $t\in\frac{\mathbb{Z}}{2}$, and $d\in\mathbb{Z}$ such that

Let $a$, $b$, and $c$ be non-negative integers. Consider the homogeneous component of order $n$ in $\operatorname{SDR}_n$ with $x$-degree $a$, $y$-degree $b$, and $z$-degree $c$. H…

Let $W$ be the Weyl group, let $w\in W$, and let $\phi$ be a character. Give $U^-_q(\mathfrak{g})$ the $\mathbb{Z}$-grading determined by $\deg(E_{\alpha_i})=1$,…