45 problems

Let ${\Gamma}$ be a connected graph, let $R$ be the polynomial ring associated to its vertices, let $I$ be the edge ideal of ${\Gamma}$, and let ${\mathfrak m}$ be the homogeneous…

Let $(R,\mathfrak m)$ be a local ring and let $I$ be an ideal of $R$. For an ideal $I$, write $\operatorname{cd}(I,R)$ for its cohomological dimension, and let $H^c_I(R)$ denote th…

Let $A$ be an integer matrix, let $S_A$ be its semigroup ring, and let $\mathfrak m$ be its homogeneous maximal ideal. For each $i<\dim(S_A)$, let $H^i_{\mathfrak m}(S_A)$ denote t…

Prime-power annihilation conjecture. For every prime power $q=p^e$, there exists $k\in\mathbb{N}$ such that

Let $R$ be a regular ring, let $\mathfrak{a}$ be an ideal of $R$, and let $H^i_\mathfrak{a}(R)$ denote its $i$th local cohomology module. Lyubeznik's conjecture. The module…

Singh's conjecture. There exists $k\in\mathbb N$ such that

Local cohomology conjecture. Under the same notation and assumptions,

Let $(R,\mathfrak{m},\mathbb{K})$ be a commutative Noetherian local ring with residue field $\mathbb{K}=R/\mathfrak{m}$, let $I\subset R$ be an ideal, and let $H_I^j(R)$ denote the…

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…

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

Suppose $X$ is embedded in a smooth variety $Y$ of dimension $d$. Choose a log resolution … of the pair $(Y,X)$ that is an isomorphism away from $X$, and let … Thus $E$ is a simple…

Let $M_{n,k}^{\mathrm{prim}}$ be the primitive quotient and let the preceding primitive vector-space conjecture identify it with $H^*(X)[\sum_{i=1}^nx_i,\sum_{i=1}^ny_i]$. Ring-str…

Generators conjecture. The space $M_{n,k}^{\mathrm{prim}}$ is spanned by products of $1/\Delta_x^{k-1}$ and polynomials of the forms $\psi_s(\mathrm{pt})$ and $\psi_s(1)$. The clai…

Let $M_{n,k}=H_L^n(O(k))^\vee$, and let $M_{n,k}^{\mathrm{shuffles}}\subset M_{n,k}$ be spanned by shuffle products $f*g$ with $f\in M_{n',k}$, $g\in M_{n-n',k}$, and $0<n'<n$. Def…

Let $L\subset\mathrm{Hilb}_n(\mathbb C^2)$ be the Lagrangian cut out by $\sum_{i=1}^n x_i^r=0$ for $r>0$, let $P$ be the Procesi bundle, let $\Omega^*$ be the bundle of differentia…

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.

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 $R$ be a Noetherian local ring of dimension $d$. A sequence $x_1,\ldots,x_d$ is a system of parameters when it generates an ideal whose radical is the maximal ideal; more gener…

Let $A$ be a polynomial ring, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Strumfels ideal, let $\operatorname{gin}(I)$ be its $\mathbb{Z}^m$-graded generic initial ide…

Let $Z$ be a reduced closed subscheme of a smooth variety $X$ of dimension $n$, let $f:Y\to X$ be a log resolution, and let $E$ be the reduced exceptional divisor over $Z$. Depth–v…

Let $X$ be a smooth variety and let $Z\subseteq X$ be a local complete intersection of pure codimension $r$. Write $F_1\mathcal{H}_Z^r(\mathscr{O}_X)$ for the first Hodge-filtratio…

Let $X$ be an $n$-dimensional scheme over $\mathbb C$ and let ${\mathfrak x}$ be a normal isolated singularity of $X$. Write $X^{\mathrm{an}}$ for the analytification and…

Let $R$ be a regular local ring containing a field, and let $I$ be an ideal of $R$. Varbaro's conjecture. If … then … This extends the known characteristic-independent cases of the…

Let $R$ be a Noetherian domain, let $S$ be a solid $R$-algebra, and let $I$ be an ideal of $R$. Solid-algebra persistence conjecture. One has … The conjecture asks whether cohomolo…