93 problems

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…

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

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 monomial conjecture. For every $t\in\mathbb{N}$,

Grothendieck's cofiniteness conjecture. If $M$ is a finitely generated $A$-module, then

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

Let $(A,{\mathfrak m},{\Bbbk})$ be a local Noetherian ring, and let ${\mathfrak b}_i$ be the ideal generated by elements $f_1,\ldots,f_i\in A$. Let $D(-)$ denote the Matlis dual. H…

Let $R$ be a commutative Noetherian ring, let $\mathfrak{a}$ be an ideal of $R$, and let $M$ be a finite $R$-module. For each integer $i\geq 0$, write…

Let $R=\mathbb Z[u,v,w,x,y,z]$, let ${\mathfrak a}$ be generated by the size-two minors of … so ${\mathfrak a}=(\Delta_1,\Delta_2,\Delta_3)R$ with $\Delta_1=vz-wy$,…

Grothendieck's conjecture. The module

Grothendieck's conjecture. For all $i$, the module

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 $(R,\mathfrak{m},K)$ be a local ring, let $I$ be an ideal of $R$, let $c$ be the cohomological dimension of $I$, and let $J=\operatorname{Ann}_R H^c_I(R)$. Lynch's conjecture.…

Let $R$ be a commutative Noetherian ring, let $\mathfrak a$ be an ideal of $R$, and let $M$ be a finitely generated $R$-module. For each integer $i$, let $H^i_{\mathfrak a}(M)$ den…

Grothendieck's conjecture. The module

Let $(R,\mathfrak m)$ be a Noetherian local ring, let $M$ be a finitely generated $R$-module of dimension $n$, and let $I$ be an ideal of $R$ such that … Here,…

Let $X$ be a variety over a field of characteristic $0$. Say that $X$ has dense $F$-injective type if, for a model of $X$ over a finitely generated $\a0\mathbb{Z}$-algebra, its red…

Let $R$ be a commutative Noetherian ring with non-zero identity, let $cmathfrak{a}$ be an ideal of $R$, and let $X$ be a finite $R$-module. Grothendieck's finiteness conjecture. Th…

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$, let $u_q=u_q(\mathfrak g)$ be the corresponding small quantum group at a root of unity satisfying the stated assumptions…

Vanishing conjecture. The local cohomology groups satisfy

Let $R$ be a standard bigraded $K$-algebra with irrelevant bigraded ideal $Q$, and let $M$ be a finitely generated bigraded $R$-module. Write $H^k_Q(M)_j$ for the component of seco…

Soit $X\subset \mathbb{P}^r_k$ un sous-schéma fermé du schéma projectif qui soit localement une intersection complète et dont toute composante irréductible soit de codimension…

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…