3 problems

Let $(A,{\mathfrak m},{\Bbbk})$ be a local Noetherian ring, and let ${\mathfrak b}_i$ be the ideal generated by elements $f_1,\ldots,f_i\in A$. Let $D(-)$ denote the Matlis dual. H…

Let $(R,\mathfrak{m},\mathbb{K})$ be a commutative Noetherian local ring with residue field $\mathbb{K}=R/\mathfrak{m}$, let $I\subset R$ be an ideal, and let $H_I^j(R)$ denote the…

Let $(R,\mathfrak m)$ be a regular local ring and let $I$ be an ideal such that $H^i_I(R)\neq 0$. Let $D(-)$ denote the Matlis dual. Lyubeznik–Yildirim conjecture. Then … The conje…