TITLE: CM for radical ideal
QUESTION [10 upvotes]: Let $R$ the polynomial ring in $n$ variables with complex coefficients and $I$ an ideal of $R$. Is it true that if $R/I$ is CM also $R/J$ is CM (where $J$ is the radical of $I$)?
Is there a relations between a resolution of $R/J$ and one of $R/I$? What if I suppose that $proj.dim(R/I)=2$?

REPLY [11 votes]: It is not true, but the example is not easy to find $I = (x_2^2-x_4x_5,x_1x_3-x_3x_4, x_3x_4-x_1x_5)$!