9 problems

Let $(R,\mathfrak{m})$ be an analytically unramified Noetherian local ring of positive dimension, and let $I$ be an $\mathfrak{m}$-primary ideal. Vasconcelos' positivity conjecture…

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 $(R,\frak{m})$ be an excellent Noetherian local ring of characteristic $p>0$ such that its completion $\widehat{R}$ is reduced and satisfies $(S_2)$. For a parameter ideal…

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $I$ be an $\mathfrak{m}$-primary ideal, and let $\mathcal{A}$ be an $I$-good filtration. Vasconcelos' uniformity conjecture.…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive dimension, and suppose that $R$ is unmixed. Let $Q$ be a parameter ideal of $R$. Vasconcelos' Vanishing Conjecture. Th…

Square-of-the-maximal-ideal conjecture. The coefficients satisfy:

Let $M$ be a finitely generated graded Gorenstein $R$-module, minimally generated by homogeneous elements of degree zero. Let $s=\operatorname{codim} M$ and…

Let $(R,\mathfrak m,\Bbbk)$ be a three-dimensional Cohen–Macaulay analytically unramified local ring, let $I$ be an $\mathfrak m$-primary ideal, and let … be the extended Rees ring…

Negativity characterization. For a parameter ideal $J$,