Hi.

I have a doubt about this fact:

Let f:XS be a flat, proper and surjective morphism of complex spaces (or locally noetherian, excellent schemes) with n-pure dimensional fibers. Then f is Cohen-Macaulay if and only if the relative canonical sheaf

$\omega^{n}{X/S}=H^{-n}(f^{!}O{S})$ is $S$-flat.

Perhaps must we add the condition : $R^{n}f_{*}\omega^{n}_{X/S}\simeq {\cal O}{S}$ ?

Thank you very much.