9 problems

Associated-prime extension conjecture. Every associated prime $P$ of $R/(I,f_1,\ldots,f_q)$ is of the form

Let $S$ be the polynomial ring in the paper, let $I\subset S$ be a polymatroidal ideal, and let $\operatorname{HS}_i(I^k)$ denote the $i$th homological shift ideal of $I^k$. Associ…

Let $G=K(n,k)$ be the Kneser graph of $k$-element subsets of an $n$-element set, and let $NI(G)$ denote its closed neighborhood ideal in the corresponding polynomial ring with homo…

Lyubeznik's conjecture. For every Lyubeznik functor $T(\,\cdot\,)$ on $R$, the module $T(R)$ has finitely many associated prime ideals.

Huneke–Lyubeznik conjecture. For every $i \ge 0$, the local cohomology module $H^i_I(R)$ has finitely many associated prime ideals.

Asymptotic associated-prime conjecture. For all $i\geq 0$,

Let $I$ be a square-free monomial ideal. Eventual stability conjecture. Then, $\operatorname{assrad} I^s$ is constant for $s\gg 0$. This is the ordinary-power analogue of the event…

Let $R$ be a Noetherian local ring, let $M$ be a finitely generated $R$-module, let $I$ be an ideal of $R$, and write ${\rm H}^i_I(M)$ for the $i$-th local cohomology module of $M$…

Let ${\mathcal I}=(I_n)_{n\ge 1}$ be an $\operatorname{Inc}$-invariant chain of monomial ideals. Associated-prime count conjecture. The number of associated primes of $I_n$ is even…