I'm a bit struck about this fact; I think it's really a silly question, but I'm not completely sure about it.
Let $X\subseteq \mathbf{P}^m$ be a projective variety; choose the best hypotheses possible, that is a noetherian separated scheme over an algebraically closed field $k$. Assume also $X$ is smooth. Then I have the following definitions:
$X$ is arithmetically Cohen-Macaulay if the coordinate ring is Cohen-Macaulay;
$X$ is (geometrically) Cohen-Macaulay if every local ring $\mathscr{O}_{X,p}$ is Cohen-Macaulay.
I know that these two definition must be different, but I can't figure out how to prove it. I know that Cohen-Macaulay is a strongly local property (i.e. a ring is Cohen-Macaulay iff every localization is) and this seems to contradict the thesis.
Could someone help with an example?