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?



Linkage and Cohen-Macaulay-ness
The question is local. So, let $R$ be a local ring which is Gorenstein. $I,J\subset R$ define $Y,Z$ as in your question. Then you have an exact sequence $0\to I\to R\to R/I\to 0$ and we are assuming that $R/I$ is Cohen-Macaulay. Notice that all $R,R/I,R/J$ have the same dimension $d$. Dualizing, one gets $0\to\omega\_{R/I}\to R\to R/J\to 0$. This implies that the depth of $R/J\geq d-1$. By going modulo a general set of $d-1$ elements in the maximal ideal, one can reduce to the case where $d=1$. Now dualize again to get, $0\to \operatorname{Hom}\_R(R/J,R)\to R\to R/I\to\operatorname{Ext}^1\_R(R/J,R)\to 0$. It is clear by naturality, that the map $R\to R/I$ is onto and thus the ext is zero. This says that depth of $R/J>0$ which is what we wanted.


