Negativity characterization of Cohen–Macaulayness via the first Hilbert coefficient

Let $(R,\mathfrak{m})$ be a Noetherian local ring that admits an embedding into a big Cohen–Macaulay module, and let $J$ be a parameter ideal of $R$. The first Hilbert coefficient $e_1(J)$ is defined by the Hilbert–Samuel polynomial of $J$.

Negativity characterization. For a parameter ideal $J$,

$$e_1(J)<0 \quad\Longleftrightarrow\quad R\text{ is not Cohen--Macaulay}.$$

This characterizes the failure of Cohen–Macaulayness through the sign of the first Hilbert coefficient in the stated class of local rings. The surrounding discussion notes that $e_1(J)=0$ for parameter ideals in Cohen–Macaulay rings and that the converse is automatic in dimension one but requires the embedding hypothesis in higher dimensions; the parser supplies no evidence that the assertion has been resolved.