10 problems

Sally's conjecture. If $R$ has almost minimal multiplicity, then $\operatorname{gr}_{\mathfrak m}(R)$ is almost Cohen–Macaulay.

Let $(A,\mathfrak{m})$ be a local ring, with $\mathfrak{m}$ its maximal ideal, and write $e_i=e_i(\mathfrak{m})$ for the Hilbert coefficients and $\lambda(A/\mathfrak{m})$ for the…

Wang's conjecture. One has

Guerrieri's conjecture. One has

Valla's conjecture. The reduction number of $\mathfrak m$ is at least two; this would imply that $\operatorname{gr}_{\mathfrak m}(R)$ is Cohen–Macaulay.

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$, let $\mu(\mathfrak m)$ denote the minimal number of generators of $\mathfrak m$, and let…

Let $(A,\mathfrak m)$ be a Noetherian local ring, and let $G(\mathfrak m)$ denote the associated graded ring of the maximal ideal filtration. Suppose that $A$ is Buchsbaum and that…

Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, let $G(A)=\bigoplus_{n\geq 0}\mathfrak{m}^n/\mathfrak{m}^{n+1}$ be its associated graded ring, and write its Hilbert series a…

Let $d$ be an integer and define the ideal … An ideal is Ratliff–Rush if it equals its Ratliff–Rush closure, and all powers $I_d^l$ are understood for positive integers $l$. Heinze…

Let $\mathbb{M}$ be a good filtration of a Cohen–Macaulay module $M$ of dimension $r$, and let $vv_j$ denote the associated deviation numbers. Huneke's conjecture. If $vv_j\le 1$ f…