I'm not sure how long this [iterative](https://mathoverflow.net/questions/43438/) [questions](https://mathoverflow.net/questions/43255/) can go on, but let me try again.  Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the coordinate ring).   Are the schematic fixed points $X^{\mathbb{G}_m}$ of $X$ Cohen-Macaulay?