794 problems

Let $\Gamma=(\mathsf{V},\mathsf{E})$ be a connected graph, let $\overline{\mathsf{E}}$ denote the relevant complementary edge set, and let $n$ be the number of vertices. Define … w…

Let $R$ be the ring under consideration, let $C$ be a semidualizing module, and let $X$ be any $R$-module. Write $\mathcal{G}_C\mathop{\rm \text{-}\dim}\nolimits X$ and…

Question. If

Let $R$ be a complete local domain containing an algebraically closed field. If the perfection of $R$ , $R_{perf}$ (union of all $p^{e}$ th roots, in the sense of positive characte…

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

For every subsemiring $S\subseteq\mathbb{R}_{\ge 0}$ with $1\in S$, if the additive monoid $(S,+)$ and the nonzero multiplicative monoid $(S\setminus\{0\},\cdot)$ are both factoria…

For every noetherian scheme $X$ of finite Krull dimension $d$, one has $K_i(X)=0$ for every integer $i<-d$.

For every commutative ring $R$, every purely-prime ideal of $R$ is purely-maximal; that is,…

Let $R$ be an admissible hypersurface, meaning that $\widehat{R}$ is a quotient of an unramified or equicharacteristic regular local ring by a nonzero element, with an isolated sin…

Core formula conjecture. One has

Exceptional-set characterization. The exceptional set $\mathcal{E}(A)$ is empty if and only if the toric ideal $I_A$ is Cohen–Macaulay.

Let $S$ be the polynomial ring and $I_{n,n}$ the special class of squarefree monomial ideals considered in the paper, and let $\operatorname{hdepth}(S/I_{n,n})$ denote the Hilbert…

Let $G$ be a graph on $n\ge 7$ non-isolated vertices. Suppose that $\operatorname{reg}(J_G)=n-1$. Projective-dimension bound. Then … The paper reports that this stronger bound is s…

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that its completion $\widehat{R}$ is reduced and satisfies Serre's $S_2$ condition. Let $B$ be a balanced big Cohen–Macaulay…

The adjusted generator bound. If

The naive generator bound. If

Ma's multiplicity conjecture.

Exact-one failure conjecture. If there is a complete intersection $J\in CI(d,d,d,d)$ that fails WLP by more than one, then there exists a complete intersection $I\in CI(d,d,d,d)$ t…

Derivation-module dimension conjecture. If

Shank–Wehlau–Broer conjecture. If $S^G$ is a direct summand of $S$ as an $S^G$-module, then $S^G$ is a polynomial ring.

Let $\Delta_{m,n}$ be a chessboard complex with $n\geq m\geq 1$, and let $S/{\mathcal F}(\Delta_{m,n})$ denote its facet ring. Regularity-depth equality conjecture. … The preceding…

Let $L_k(n_1,n_2,\ldots,n_k)$ be the lattice considered in the paper, and let $I_{L_k(n_1,n_2,\ldots,n_k)}$ denote its join-meet ideal in the associated polynomial ring. Crystal co…

Polynomial Betti-bound conjecture. All Betti numbers of $S/I$ are bounded, as functions of $d$, by polynomials of degree at most $p-1$.

Virtual-resolution minimality conjecture. The virtual resolution of $M(\mathbf d)$ from Proposition is the minimal free resolution of $M_{\geq\mathbf d}(\mathbf d)$.

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…