Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?



Irreducible component of a Cohen-Macaulay variety
Start with your favorite example of an affine irreducible variety $X$ that is not Cohen-Macaulay. Embed $X$ in $\mathbb A^N$, and call $c$ its codimension. Now take $c$ general polynomials that contain $X$: their intersection $Y$ has codimension $c$, and contains $X$. Then $Y$ is a complete intersection (hence Cohen-Macaulay) subvariety of $\mathbb A^N$, and contains $X$ as an irreducible component.


