34 problems

Let $A$ be an artinian Gorenstein algebra of socle degree $d\geq 3$ whose defining ideal is generated by quadrics. The weak Lefschetz property means that multiplication by a genera…

Let $\Lambda$ be an Artin algebra, and write $\operatorname{id}_\Lambda\Lambda$ for its left self-injective dimension and $\operatorname{id}\Lambda_\Lambda$ for its right self-inje…

Let $R$ be an Artin algebra. Its left self-injective dimension and right self-injective dimension are the corresponding self-injective dimensions of $R$. The Gorenstein Symmetry Co…

Realizability conjecture. All potential rank matrices for codimension-three Gorenstein sequences are actually realizable.

Let $A=R/I$ be a generic Artinian Gorenstein $K$-algebra of odd socle degree $s=2t-1\geqslant 3$ and embedding dimension $n\geqslant 4$. The ideal $I$ is the homogeneous defining i…

SI-sequence conjecture. Every Gorenstein $h$-vector in codimension $r=4$ is an SI-sequence.

Failure conjecture. Not all height four Gorenstein sequences satisfy the condition that $\Delta H_{\leq j/2}$ is an $O$-sequence.

Stanley–Iarrobino conjecture. Every Gorenstein $h$-vector, in every codimension $r$, is an SI-sequence.

Let $R=k[x_1,\ldots,x_n]$, let ${\mathfrak c}\subset R$ be a general complete intersection of type $(d_1,\ldots,d_c)$, and let $A=R/I$ be relatively compressed with respect to…

Let $G$ be a simple group with maximal compact subgroup $G_c$, let ${\fam\Bbbfam V}_\lambda$ be a weight multiplicity free representation, and let…

Compactly Gorenstein algebra conjecture. All Artin algebras should be compactly Gorenstein.

Let $\Lambda$ be an Artin algebra. The Auslander Gorenstein conjecture. If $\Lambda$ satisfies the Auslander condition, meaning that it is $n$-Gorenstein for every positive integer…

Let $A$ be a Gorenstein local algebra of dimension $n+1$ with maximal ideal $\mathfrak{m}$, and let $U$ be a hyperplane in $\mathfrak{m}$ complementary to $\operatorname{Soc} A$. B…

Minimal-generator conjecture. If $d$ is odd, then $\operatorname{Ann}(H_{2d+1})$ has no minimal generators in degree $d+2$, and

Let $A=k[\![x,y]\!]/J$ be an AG $k$-algebra, let $I$ be an ideal in $A$, and let $H$ be a $2$-admissible $G(I)$-Gorenstein sequence. A symmetric decomposition is a decomposition of…

Let $A=k[ … /J$ be an AG $k$-algebra. Let $H=(h0,h1,ldots,hu)$ be a $2$-admissible sequence, where a sequence is partially strict unimodal if…

Koszul Roller Coaster conjecture. There exists such an algebra satisfying

Let $S={\Bbbk}[x,y,z]$, let $d$ be a positive integer, and let $[S/I]\in\mathrm{Hilb}^d(\mathbb{A}^3)$ be a smoothable point. Broken Gorenstein characterization conjecture. The fol…

A finite scheme on a smooth threefold is said to have a broken Gorenstein structure when it admits the broken Gorenstein structure introduced in the paper. Exhaustivity conjecture.…

Let $R$ be an Artin algebra. Say that $R$ is left weakly Gorenstein or right weakly Gorenstein according to the corresponding one-sided condition. The Weakly-Gorenstein Symmetry Co…

Realizability conjecture. Every potential JDT arising for $s=6$ is realized by some pair $(A,\ell)$.

WLP conjecture. Every Artinian Gorenstein quotient of $k[x,y,z]$ has the WLP.

Additive-action multiplicity conjecture. There are at least two non-equivalent induced additive actions on $X$.

Iwanaga–Gorenstein Wakamatsu tilting conjecture. Any Wakamatsu tilting $\Lambda$-module is tilting.

Self-orthogonal module conjecture. Every self-orthogonal $\Lambda$-module has finite projective dimension.