3 problems

Let $R={\Bbbk}[x_1,\dots,x_r]/J$, where $J\subseteq{\Bbbk}[x_1,\dots,x_r]$ is homogeneous, and suppose $\operatorname{char}{\Bbbk}=p>0$. Let $I\subseteq R$ be an ideal, and conside…

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$, and let $I$ be an $\mathfrak m$-primary ideal. The ideal $I$ is stable if it satisfies $I^2=QI$ for a minimal…

Let $R=\mathbb{K}[X_1,\ldots,X_n]$ be a standard graded polynomial ring over a field $\mathbb{K}$ of characteristic $p$. Let $I\subset R$ and $J\subset R$ be homogeneous ideals, an…