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?

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 $proj.dim(R/I)=2$?

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)?

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?