1,519 problems

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

Auslander–Reiten conjecture. If

Let $S$ be a numerical semigroup. Write $\operatorname{e}(S)$ for its embedding dimension, $\operatorname{n}(S)$ for the number of elements of $S$ below its conductor, and…

Let $S=\mathsf k[x_1,\ldots,x_n]$ be a polynomial ring, and let $R=S/I$ be a general quotient of type $(N;d_1,\ldots,d_N)$: the ideal $I$ is minimally generated by forms…

Let $\Gamma$ be a symmetric numerical semigroup, let $\Bbbk$ be a field, and let $R=\Bbbk[\Gamma]$ be its semigroup algebra. Let $M$ be a 2-generated ideal of $R$, viewed as an $R$…

Eisenbud–Green–Harris conjecture. The ideal $I$ has the same Hilbert function as an ideal containing $x_1^{a_1},\dots,x_n^{a_n}$. This conjecture predicts that the Hilbert function…

Let $(R,\mathfrak m,k)$ be a Noetherian local ring with a canonical module $K_R$. Tachikawa's conjecture. If … for all $i>0$, then $R$ is Gorenstein. This is the commutative versio…

Let $p$ be an odd prime and let … which is a singular ring of characteristic $p$ and dimension $d-1$. For a Noetherian local ring $(R,\mathfrak{m})$ of characteristic $p$, write…

Let $R$ be a commutative Noetherian ring such that $\operatorname{Spec}(R)$ is connected, and let $M$ be a non-zero, finitely generated $R$-module of finite projective dimension. F…

Strong Lefschetz conjecture. For any smooth $F\in S_d$, any integer $k\in[0,T/2)$, and any generic linear form $\ell\in S_1$, the induced multiplication map is an isomorphism. The…

Let $G$ be a finite simple graph, and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring associated to $G$. Write $\mathcal{R}(I_c(G))$ for the Rees algebra of…

Huneke's conjecture. The Bass numbers $\mu^i(\mathfrak{p},H^j_{\mathfrak{a}}(R))$ are finite for all $i,j\in\mathbb{N}_0$ and every $\mathfrak{p}\in\operatorname{Spec}(R)$. This co…

Vasconcelos's differential conjecture. If

Let $\mathbb{k}$ be a field, let $R=\mathbb{k}[x_1,\ldots,x_d]$, and let $I\subseteq R$ be a radical homogeneous ideal of bigheight $h$. Harbourne's conjecture. One has … for every…

Let $A=S/I$ be a Cohen–Macaulay quotient ring of codimension $c$. In each homological term of its minimal free resolution, let $a_i$ be the maximal twist and $b_i$ the minimal twis…

Let $G$ be a finite simple graph, with $n(G)$ denoting its number of vertices and $J_G$ its binomial edge ideal in $S=K[x_i,y_i:i\in V(G)]$. Matsuda–Murai's conjecture. … if and on…

Sturmfels–Sullivant's conjecture. The following statements are equivalent:

Let $S=k[X_0,\dots,X_n]$ be a polynomial ring over a field $k$, let $I\subseteq S$ be an ideal, and let $<$ be a monomial order such that the initial ideal $\operatorname{in}_<(I)$…

Lyubeznik's conjecture. For each $i\geq 0$, the set $\operatorname{Ass}_R H^i_I(R)$ is finite.

Let $(R, \mathfrak{m}_R, k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero, and let $\Omega_R$ denote its universally finite module of…

Let $(R,\mathfrak{m})$ be a Noetherian local ring or a standard graded $K$-algebra, and let $I\subset R$ be an ideal, homogeneous when $R$ is a $K$-algebra. The depth-function conj…

Hochster's conjecture. The invariant ring of any linearly reductive group acting on a polynomial ring over a field is Cohen–Macaulay.

Hochster's monomial conjecture. For every $t\in\mathbb{N}$,

Let $S=k[x_1,[...truncated]$ be a polynomial ring, and let $S^{(c)}$ denote its $c$-th Veronese subring. In zero characteristic, the Green–Lazarsfeld index satisfies … For $n=3$, O…

Auslander's conjecture. Finite-dimensional modules over a finite-dimensional $k$-algebra satisfy (AC).