# Duality and isomorphism of functor

Hi.

First of all thanks to Zsolt for the answer to the question on "Cohen Macaulay morphism".

I want to show for which proper and flat morphisms $f:X\rightarrow S$ of complex spaces with $n$ pure dimensional fibers (or locally noetherian, excellent schemes with n-equidimensional fibers) do the two functors $G\rightarrow f^{*}G\otimes \omega^{n}_{X/S}$ and $G\rightarrow H^{-n}(f^{!}G)$ agree?

If $f$ is a Cohen Macaulay morphism, it is obviously true. But I don't know if that is the only case for which the two functors agree...

Partial answer: If $f$ is flat, proper, surjective $S_{2}$-morphism with fibers without embedded components and $\omega^{n}_{X/S}$- $S$-flat, then the two functor agree as well.

Questions:

1) Is there a morphism, other than a CM-morphism, which satisfies these assumptions?

2) Can we relax the hypothesis on the fibers?

Thank you very much.

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Can you give a hint on how one proves the partial answer you gave? I would be interested. Thanks. –  Zsolt Patakfalvi Jun 30 '10 at 12:58
Could you define $\omega^{n}_{X/S}$? –  Sándor Kovács Oct 24 '10 at 17:20

The partial answer for $f$ flat, proper, surjective $S_{2}$-morphism with fibers without embedded components and $\omega^{n}_{X/S}$- $S$-flat is due to the fact that the set of points which are not Cohen-Macaulay is of codimension $\geq 2$...