27 problems

Let $N\geq 1$. Write $M(\Gamma_0(N),\mathbb{Z}[\frac{1}{6N}])$ for the graded algebra of modular forms for $\Gamma_0(N)$ with coefficients in $\mathbb{Z}[\frac{1}{6N}]$. Weight-six…

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…

Hirose–Watanabe–Yoshida conjecture. One always has

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 $E$ and $F$ be finite graphs, and let $K$ be a field. For a … -graded case with its natural $\mathbb{Z}[x,x^{-1}]$-module structure. Let $T_E$ denote the talented monoid of $E$…

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…

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 $R=\bigoplus_{n\in G}R_n$ 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 $R=\bigoplus\limits_{n\in G}R_n$ 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 $R$,

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 $S$ 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…

Graded-ring generalization of Kaplansky's conjectures. Every unit in $R$ is homogeneous, $R$ is a domain, and every idempotent in $R$ is trivial.

Let $R$ be a standard graded strongly $F$-regular ring, with $R_0$ an $F$-finite field of characteristic $p>0$. Let $\mathfrak{m}$ be the unique homogeneous maximal ideal of $R$. H…

Conjecture on log-canonical thresholds and a-invariants. 1. …

Let $D$ be a … , supported at points $P_1,,P_n$, and let $S_D$ be its canonical ring. A minimal relation means a relation in a minimal presentation of $S_D$ by generators. The poin…

Homological-degree regularity conjecture. There exists a polynomial $f(t)$ of degree $d$ such that

Let $R$ be a standard graded Gorenstein ring of countable Cohen–Macaulay type, meaning that it has only countably many indecomposable graded Cohen–Macaulay modules up to shifts. Hy…

Base conjecture. Every graded algebra retract of $S/I$ has a base.

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$. Let $I$ be a homogeneous ideal containing a regular sequence $f_1,\dots,f_n$ with degrees … A degree-$a_i$ form is s…

Hazrat's strong classification conjecture. The functor $K_0^{\operatorname{gr}}$ is fully faithful from the category of unital Leavitt path algebras with graded homomorphisms modul…