14 problems

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

Finiteness conjecture. The module $H^i_I(M)$ has finitely many associated primes for every $i$.

Let $(A,\mathfrak m)$ be a local Noetherian ring, let $J$ be an ideal of $A$, and let $M$ be a finitely generated $A$-module. Write $H^i_J(M)$ for the local cohomology modules of…

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…

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 $G$ be a graph, and let $J(G)$ denote its edge ideal. Francisco–Ha–Van Tuyl conjecture. The sets $\operatorname{ass}(J(G)^s)$ are increasing in $s$; equivalently,…

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 ring and $I\subset R$ a proper ideal. Write … which is well-defined by Brodmann's stability theorem. Ratliff's conjecture. One has … The conjecture concerns…

Let $R$ be a weakly Laskerian ring. Write ${\rm Min}(0)$ for the set of minimal prime ideals of the zero ideal and $\operatorname{Ass}_R R$ for the associated-prime set of the $R$-…

Let $(R,\frak{m})$ be a local ring and let $M$ be an $R$-module. A coassociated prime ideal of $M$ is called minimal if it is minimal, under inclusion, among the coassociated prime…

Let $I$ be a square-free monomial ideal, and suppose that $I^n=I^{(n)}$ for every $n\leq N$, where … Here $d$ is the number of variables of the polynomial ring containing $I$, and…