39 problems

Hirose–Watanabe–Yoshida conjecture. One always has

Let $R=\bigoplus_{n\in G}R_n$ be a ring graded by a torsion-free group $G$, and let $J(R)$ denote its Jacobson radical. An ideal is graded when it is the direct sum of its homogene…

Let $E$ be a directed graph and let $L_K(E)$ be its Leavitt path algebra with its natural grading. The talented monoid of $L_K(E)$ is the graded monoid associated with the graded f…

Let $R$ be the graded section ring in the preceding construction, and let $v_3,v_4,v_5,v_6$ denote the generators of the module in the indicated weights $(1,4)$ and $(1,5)$. Degree…

Beheshti–Eisenbud conjecture. For $q\gg0$,

Let $R$ be a graded Noetherian ring, and let $\mathcal{D}(R)$ be the relevant derived or stable category with sphere object $S$. Write $\mathcal{GH}$ for the generating hypothesis,…

Let $R$ be the graded or local two-dimensional regular ring considered in the paper, let $I$ be a contracted ideal, let $h_i(I)$ denote the coefficients of the $h$-vector of $I$, a…

Let $R$ be a Cohen--Macaulay ring, let $\mathfrak{p}$ be a prime ideal of finite projective dimension, and let $\mathrm G=\bigoplus_{t\geq 0}\mathfrak{p}^t/\mathfrak{p}^{t+1}$ be t…

Negative-invariant conjecture. If $\operatorname{char}(k)=0$ and $b(R)<0$, then $R$ is of strongly F-regular type. If, moreover, $R$ is $\mathbb Q$-Gorenstein, meaning that…

Let $A$ and $B$ be the graded rings and $R=A\times_T B$ the graded fiber product considered above, with canonical modules $\omega_R$ and $\omega_{A,B}:=\omega_A\oplus\omega_B$. Ass…

Let $S$ be a nonstandard graded polynomial ring, and for a positive integer $e$ let $S^{(e)}$ denote its $e$-th Veronese subring. A graded algebra is nonstandard Koszul when it has…

The paper considers a polynomial ring $S$ with an arbitrary $\mathbb Z$-grading and its truncations $S_{\geq d}$, whose homological properties are studied through their graded free…

Let $R$ be a standard graded algebra, and let $d>0$. A pair $(\ell,Q)$ of exact zero divisors means that $\ell$ is a general linear form, $Q$ is an element of degree $d-1$, and the…

Let $R=\bigoplus\limits_{n\in G}R_n$ be a ring graded by an Abelian group $G$, and let $H$ be the torsion subgroup of $G$. An idempotent is an element $e$ satisfying $e^2=e$. The s…

Let $R=\bigoplus\limits_{n\in G}R_n$ be a ring graded by a torsion-free group $G$, where $0$ denotes the identity element of $G$. An idempotent is an element $e$ satisfying $e^2=e$…

Let $R=bigoplus{n i G}Rn$ be a ring graded by a torsion-free group $G$, and let $I$ be a left ideal of $R$. The ideal $I$ is unfaithful when its annihilator is nonzero. **The g…

Graded-ring conjecture. For any ideal $\alpha$ of a graded ring $R$,

Strong Graded Classification Conjecture. The functor

Graded Classification Conjecture. The following conditions are equivalent:

Let $Y$ be the surface in the preceding construction, with divisors $F$ and $S$, and let … be its bigraded section ring. Let $u_0,u_1,v_1,v_2$ be sections of $\mathcal O(2F)$,…

Folklore finite-generation conjecture. The graded ring $J_{*,L,*}^w$ is finitely generated over $\mathbb{C}$ for every positive-definite integral lattice $L$.

Reduction to $k+1$ variables conjecture. For a generic linear form $lin A_1$, either $bar W$ contains no power of a linear form, or

Canonical-module self-duality conjecture. If $\omega_R$ has analytic spread $1$, then some power of $\omega_R$, regarded as a fractional ideal, is $\omega_R$-self-dual up to a shif…

Let $K(5)$ denote the paramodular group of level $5$, and let … be its graded ring of paramodular forms. Marschner's conjecture. The graded ring $M_*(K(5))$ is Cohen–Macaulay. The…

Let $S$ be the bihomogeneous polynomial ring and let $\overline{S}$ denote the quotient defined in the paper. Fix $\mathbf n=(n_1,n_2)\in \mathbb{Z}^2_{\geq 1}$, set…