difference between Cohen Macaulay and locally Cohen Macaulay curve

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.

The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in particular) $X$ is Cohen Macaulay if for every $x \in X$, there is an open subset $U$ of $X$ containing $x$ such that the module $\mathcal{O}_{X}(U)$ is Cohen Macaulay.

On the other hand, although there is a lot of papers on locally Cohen Macaulay curves, I was not able to find the precise definition for it. It seems that a curve $C$ is locally Cohen Macaulay if it has no embedded nor isolated points. (As Harthorne says in his paper "Stable reflexive sheaves" (https://link.springer.com/article/10.1007%2FBF01467074) for instance).

So my questions are: 1) Where can I find the precise definition of locally Cohen Macaulay curve? 2) If the definition of locally Cohen Macaulay curve that I stated is correct, how can I see that Cohen Macaulay implies locally Cohen Macaulay? (Which I believe that should be the case because of their names)

• I think that, for a locally noetherian scheme $X$, "locally Cohen Macaulay" simply means that $\mathcal{O}_{X, x}$ is a Cohen Macaulay local ring for all $x \in X$. Commented Sep 7, 2018 at 15:29
• @FrancescoPolizzi - if that's the definition, then the two would seem to be equivalent? Because $\mathcal{O}_X(U)$ is CM, by definition, if every localization at a prime is CM. Commented Sep 7, 2018 at 15:34
• Yes, they are. I will post an answer. Commented Sep 7, 2018 at 15:35

Following the usual meaning of "local" in Algebraic Geometry, a locally noetherian scheme is locally Cohen-Macaulay if and only if $\mathcal{O}_{X, \, x}$ is a Cohen-Macaulay local ring for every $x \in X$.
Moreover, a locally noetherian scheme of dimension $\leq 1$ is Cohen-Macaulay if and only if it has no embedded points, see [The Stacks Project, Lemma 30.4.4].