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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)

Thanks in advance.

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    $\begingroup$ 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 $x \in X$. $\endgroup$
    – Francesco Polizzi
    Commented Sep 7, 2018 at 15:29
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    $\begingroup$ @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. $\endgroup$
    – benblumsmith
    Commented Sep 7, 2018 at 15:34
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    $\begingroup$ Yes, they are. I will post an answer. $\endgroup$
    – Francesco Polizzi
    Commented Sep 7, 2018 at 15:35

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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 $x \in X$.

This is actually equivalent to your definition of Cohen-Macaulay scheme, see [The Stacks Project, Lemma 27.8.2].

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].

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    $\begingroup$ I think it is dangerous to say that this meaning of "local" is the usual one in algebraic geometry, and your mention of local noetherianness illustrates my point. Unfortunately, there are two usual meanings; it might be less confusing if one of them would be called pointwise. $\endgroup$
    – Fred Rohrer
    Commented Sep 7, 2018 at 18:47

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