128 problems

Artin's conjecture. Birationally, there is a short list of noncommutative projective surfaces, with the generic case being a Sklyanin algebra.

Bahturin–Regev's conjecture. The regular quantum commutative decomposition of $R$ is minimal if and only if

Let $A$ be a connected regular algebra of exponential growth, where regularity means finite global dimension together with the stated Gorenstein property. Bondal's coherence conjec…

Anick's conjecture. Suppose $A$ is an augmented connected graded $K$-algebra with polynomial growth and with global dimension $d<\infty$. Then the Hilbert series of $A$ is given by…

Bahturin–Zaicev's conjecture. The inequality

Watanabe's Bold Conjecture. There is another standard graded Artinian complete intersection $B$ of the same socle degree $d$, cut out by quadrics, together with an injective algebr…

Goncharov's direct sum conjecture. The algebra $\mathrm{Z}$ is graded by weights; equivalently, the weight decomposition is a direct sum.

Let $A$ be a normal graded algebra whose associated graded singularity is isolated, and consider the homogeneous $k$-linear derivations of $A$ with respect to a positive grading. W…

Scalewise birational extension conjecture. There exists an ideal $H\subset\hat R$ with $H\cap R=(0)$ and a valuation $\hat\nu_-$ centered at $\hat R/H$ such that

Let $P_{BF}^{(1,1)}$ be the relative parabose set generated by $b^{\pm}$ and $f^{\pm}$. Consider representations on a complex, infinite-dimensional pre-Hilbert space with adjointne…

Let $R$ code a permutation group with no finite orbits, and let its age algebra over $\mathbb{C}$ be the associated graded algebra of finite relational structures up to isomorphism…

Quadratic Koszul reconstruction conjecture. If the relations of $A$ are purely quadratic, then dualising the Yoneda product should give a presentation of $A$.

A graded algebra $A$ is regular if it has finite global dimension $d$ and satisfies the Gorenstein condition … The algebras of polynomial growth are called Artin–Schelter (AS) regu…

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 $R/I$ be a Cohen–Macaulay standard graded algebra of codimension $r$ with minimal graded free resolution, and define $m_i$ and $M_i$ as the least and greatest shifts in homolog…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $I\subset R$ be a graded ideal. For a minimal graded free resolution of $R/I$, let $M_i$ denote…

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 $S$ be a strongly noetherian connected graded $k$-algebra, let $P$ be a finitely generated graded $S$-module, and let $h$ be a Hilbert function. Suppose that $S$ satisfies the…

Dimension conjecture. In both the finite-cohomological-dimension theorem and the global-dimension corollary, the correct dimension should be $\dim X$; in particular, the relevant u…

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…

Vanishing conjecture. For $N\geq 3$, any product of more than $N$ one-forms $d\xi^k$ must vanish.

Let $S$ be a finitely generated graded $\Bbbk$-algebra that is Koszul, and let $T^i_{S/\Bbbk}$ denote its $i$th cotangent cohomology, with its internal grading. The extremal-weight…