Abstract
Let $(R,\mathfrak{m} )$ be an unmixed Noetherian local ring, $Q$ a parameter ideal and $K$ an $\mathfrak{m} $-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and $g_1(Q)$, the Hilbert coefficients of $Q$ with respect to $K$. As a consequence, we obtain a result of Ghezzi, et al., which settles the negativity conjecture of Vasconcelos {vanishing-conjecture} in unmixed local rings.
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