822 problems

For $1\leq i<j\leq n$, let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$, let $\mathcal{S}_{ij}^2$ be its square image, let $P$ be the associated symmetric matrix,…

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…

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

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…

Eisenbud–Green–Harris conjecture. The ideal $I$ has the same Hilbert function as an ideal containing $x_1^{a_1},\dots,x_n^{a_n}$. This conjecture predicts that the Hilbert function…

A small Cohen–Macaulay module over a local ring is a finitely generated module whose depth equals the dimension of the ring. The small Cohen–Macaulay conjecture. Every complete loc…

Let $R$ be an algebra over a field $K$, let $z$ be an indeterminate, and let $x_1,\dots,x_n$ be commuting indeterminates. Zariski's cancellation conjecture. If … then … The conject…

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

Let $k$ be a field of characteristic $0$, let $n\ge 1$, and let $f\in (x_1,\ldots,x_n)^2\subset k[[x_1,\ldots,x_n]]$ define an isolated critical point at the origin. If…

Let $m,n\geq 1$, let $\lambda=(n-k,1^k)$ be a hook partition of $n$ with $0\leq k\leq n-1$, and let $\mathcal{R}_{m,n}=\mathbb{C}[x_{r,j}:1\leq r\leq m,\ 1\leq j\leq n]$ carry the…

For every integer $N\ge 2$ and every integer $s\ge 1$, let $I\subseteq k[x_0,\ldots,x_N]$ be the homogeneous defining ideal of a general set of $s$ points in $\mathbb{P}^N$ over an…

For the Hermite normal form simplices $S_{\boldsymbol a}=\operatorname{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol a)$, where $\boldsymbol a=(a_1,\ldots,a_{d-1},N)$, and…

Does every commutative Bézout domain $R$ have the property that every matrix $A\in R^{m\times n}$ admits a Smith normal form; that is, do there exist invertible matrices…

For every numerical semigroup $H=\langle a_1,\ldots,a_n\rangle$ minimally generated by $n$ elements, let $I_H=\ker\bigl(k[x_1,\ldots,x_n]\to k[t]\bigr)$, where $x_i\mapsto t^{a_i}$…

Let $(R,\mathfrak m,k)$ be an $F$-finite, unmixed, $d$-dimensional Noetherian local ring of characteristic $p>0$. If $R$ is not regular, then its Hilbert–Kunz multiplicity satisfie…

Let $\Delta$ be a pure vertex-decomposable simplicial complex on the vertex set $[n]$, and let $I_{\Delta^{\vee}}$ denote the Stanley–Reisner ideal of its Alexander dual. The Lu–Wa…

Let $\mathbf{k}$ be an algebraically closed field of characteristic zero, and let $A$ be a finite-dimensional local $\mathbf{k}$-algebra with $\dim_{\mathbf{k}} A=\ell$. Is it true…

Let $A$ be a finite-dimensional algebra over a field. Assume that $A$ satisfies the Auslander condition: for every finitely generated left $A$-module $M$, there exists an integer…

For every finite bounded lattice $L$ and integers $a,b$, if $\widetilde{H}_{a+b-2}\!\left(\Delta\left(L\setminus\{\hat{0},\hat{1}\}\right);\Bbbk\right)\neq 0$, then there exist com…

Let $k$ be a field, let $S=k[x_1,\ldots,x_n]$, and let $I\subseteq S$ be the homogeneous ideal of $n$ general points in $\mathbb{P}^{n-1}$. For every integer $m\geq 1$, determine e…

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup minimally generated by $a_1,\ldots,a_n$, let $S=k[x_1,\ldots,x_n]$, and let…

For every finite matroid $M$ of rank $r$, let $\Delta(M)$ be its independence complex and let $h(M)=(h_0,\ldots,h_r)$ be defined by…

For every complex hyperplane arrangement $\mathcal{A}$, if its Orlik–Solomon algebra $\operatorname{OS}(\mathcal{A})$ is Koszul, then the intersection lattice $L(\mathcal{A})$ is s…