Vasconcelos' Vanishing 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. Then

$$\mathrm{e}_{1}(Q)=0$$

if and only if $R$ is Cohen–Macaulay. The conjecture asks whether vanishing of the first Hilbert coefficient characterizes Cohen–Macaulayness among unmixed local rings; the source states that it was subsequently settled.