# How to show that the coordinate ring of a finite set of points in projective space is Cohen-Macaulay?

Could anyone give me a hint how to show this one:

Let $V$ be a finite set of points in projective space. How to show that the coordinate ring of $V$ is Cohen-Macaulay?

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Hint: For a $k$-dimensional ring, Cohen-Macaulay is equivalent to $S_k$. What is $k$ here, and what does $S_k$ mean explicitly? –  David Speyer May 14 '12 at 12:18
Just to check, when you say coordinate ring, you mean that you have $X \subset \mathbb{P}^n$ and you are looking at the corresponding homogenous quotient of $\mathbb{C}[x_0, \ldots, x_n]$, right? @Georges Elencwajg's hint assumes that you mean $H^0(X, \mathcal{O})$, which is also a reasonable interpretation, but makes this question very easy, so I don't think it's what you mean. –  David Speyer May 14 '12 at 12:20

Proof 1: Regular $\implies$ Cohen-Macaulay.
Proof 2: Artinian $\implies$ Cohen-Macaulay.