Suppose I have a reduced l.c.i. scheme with two irreducible components: $X = Y \cup Z$. I want to say that if $Y$ is Cohen-Macaulay then $Z$ is as well.

I think this follows from Eisenbund Theorem 21.23 (which has a typo: the first $J = (0:_A I)$ should be deleted). Or from Peskine and Szpiro, "Liaison des variétés algébriques," Proposition 1.3, which is essentially the same.

Am I understanding correctly?