11 problems

Let $\frak{a}$ be an ideal in $\mathbb{Z}[X_1,\ldots,X_n]$, contained in $(X_1,\ldots,X_n)$, and let $\frak{a}_p$ be the localization at $(X_1,\ldots,X_n)$ of its image in…

Converse conjecture. If $I$ is symbolic $F$-split, then

Let $(R,\mathfrak m)$ be a noetherian local ring of prime characteristic. Let $f_1,f_2,\ldots,f_r$ be part of a system of parameters, let $J$ be an ideal generated by a full system…

Let $(R,\mathfrak m)$ be a noetherian local ring of prime characteristic. Let $f_1,f_2,\ldots,f_r$ be part of a system of parameters, let $J$ be an $\mathfrak m$-primary ideal, and…

Let $(R,\mathfrak m)$ be a noetherian local ring of dimension $d$ containing a positive characteristic field. Let $J$ be an ideal generated by a full system of parameters, and let…

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.…

Let $(R,\mathfrak{m},K)$ be a local ring. Let $I\subseteq R$ be an ideal generated by a part of a system of parameters $(f_1,\ldots,f_\ell)$, and set $\overline{R}=R/I$. Let $J$ be…

Let $(R,\mathfrak m)$ be a $d$-dimensional Noetherian local ring of characteristic $p>0$. Let $J\subseteq\mathfrak m$ be an ideal generated by a system of parameters, and let…

Let $(R,\mathfrak{m})$ be a $d$-dimensional Noetherian local ring of characteristic $p>0$. Let $J\subseteq\mathfrak{m}$ be an ideal generated by a full system of parameters, and le…

F-threshold multiplicity conjecture. One should have