3 problems

Let $(R,\mathfrak{m})$ be a Noetherian local ring or a standard graded $K$-algebra. Let $I\subset R$ be an ideal, homogeneous when $R$ is a $K$-algebra. The Serre-depth stabilizati…

Formal-grade conjecture. The Cohen–Macaulay condition in $(\ast)$ can be replaced by a weaker condition when the family of ideals is suitably restricted.

Excellent-set conjecture. If $A\subseteq\Delta$ is excellent and $\Delta$ satisfies $(S_{\ell})$, then $\Delta_A$ satisfies $(S_{\ell})$.