13 problems

Let $\Lambda$ be a monomial triangular algebra, meaning a triangular algebra whose defining relations are paths. Its Hochschild cohomology is denoted by…

Let $S$ be a polynomial ring in $n$ variables, let $I\sim\mathcal{I}(n,D,p)$, and consider the random Artinian algebra … where $n\geq 3$. Set $p=1/D^2$. The vanishing WLP probabili…

Let $I$ be a random monomial ideal in the Erdős–Rényi model considered in the paper, and let $S/I$ be its associated algebra with Hilbert function $h_{S/I}$. Based on simulations,…

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring, let … where $1\leq a_1\leq a_2\leq\cdots\leq a_n$, and let ${\mathfrak n}$ be the graded maximal ideal of $R$. Let ${\mathcal P}$ be…

Let $a\geq b\geq c$, set $d=a+b+c$, and define … The weak Lefschetz failure conjecture. The algebra $R_{a,b,c}$ fails the weak Lefschetz property if and only if both of the followi…

SLP classification conjecture. The equigenerated monomial almost complete intersections of codimension three with the SLP are precisely those given by

Let $\mathbb{k}[x_1,\ldots,x_n]$ be a polynomial ring. For a positive integer $u$ with $\binom{n}{2}<u\leq\binom{n+1}{2}$, let $\operatorname{Lex}(u)$ and…

Let $\mathbb{k}[x_1,\ldots,x_n]$ be a polynomial ring, and let $u$ degree-two forms generate a subalgebra. For a positive integer $u$ with $\binom{n}{2}<u\leq\binom{n+1}{2}$, let…

Characteristic bound conjecture. Then $R/I$ has the weak Lefschetz property in every characteristic $p>\frac{1}{2}(a+b+c)$.

The odd axial-puncture criterion.

Let $k[X_1,\ldots,X_n]$ be a polynomial ring over a field $k$, and let $R$ be a normal monomial subalgebra of this ring. A finitely generated projective $R$-module is a finitely ge…

Vanishing conjecture. Based on computer evidence, $\operatorname{Koszul}(n,N,p)$ is zero when

Smoktunowicz–Bell conjecture. Every prime monomial algebra $A$ is either PI, primitive, or has nonzero Jacobson radical.