12 problems

Let ${\Bbb K}$ be a field, let $q$ be a nonzero element of ${\Bbb K}$, and let $R_t({\bf X})={\Bbb K}_q[{\bf X}]/{\cal I}_t({\bf X})$ be the quantum determinantal ring, where…

Depth consequence. Each Tor group occurring in the conjectured isomorphism has depth at least $2$ at $J$.

Maximal Cohen–Macaulay conjecture. For every integer $i$ with $1\le i\le c+p-1$, the Schur power $S_i(\bigwedge^pM)$ is a maximal Cohen–Macaulay $R/I_{t+1-p}$-module.

Tor symmetry conjecture. One should have

Buchsbaum–Eisenbud conjectural resolution. A resolution of $\operatorname{coker}(\varphi^*_{p-1,1})$ should always start out as

Let $R=S/I_t(X)$ be the determinantal ring associated with an $n\times n$ matrix of indeterminates, and let $c^{\mathfrak{m}}(\mathfrak{m})$ denote the F-threshold of its homogeneo…

Let $R=S/I_t(X)$, where $X$ is an $n\times m$ matrix of indeterminates, $t=m\leq n$, and $c^{\mathfrak{m}}(\mathfrak{m})$ denotes the F-threshold of the homogeneous maximal ideal.…

The determinantal-ring conjecture. If $0\neq[M]\in\operatorname{Cl}(R)$, then $M$ is exactly $(m+n-2t+1)$-semidualizing. Hence

Let $X=(x_{ij})$ be an $m\times n$ matrix of indeterminates over a field $K$ of characteristic $0$, let $R=K[X]$, and let $A_t\subseteq R$ be the $K$-subalgebra generated by the…

Let $(\gamma|\lambda)$ be a bi-shape occurring in $(S_t)_\mu$. Suppose that there exists a $1$-predecessor $\mu'$ of $\mu$ containing a $t$-bi-predecessor $(\alpha|\beta)$ of…

Let $S_t(m,n)$ be the algebra generated by the $t$-minors of a generic $m\times n$ matrix, let $J_t(m,n)$ be its ideal of relations, and let ${\bf f}_{u,v}$, ${\bf g}_u$,…

Let $\mathbb{K}$ be a field of prime characteristic $p$, let $R=\mathbb{K}[\![x,y,z,u,v,w]\!]$, and let $I$ be the ideal generated by the $2\times 2$ minors of the matrix … Set…