6 problems

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 $(\gamma|\lambda)$ 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 $\mathbb{K}$ 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…