2 More grammar

Hi.

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

I want to show for which morphism $f:X\rightarrow S$ 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 functor 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 dont I don't know if that is the only case for which the two functor 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

Questions:

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

2) Can we relax the hypothesis on the fibers?

Thank you very much.

1

# 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 morphism $f:X\rightarrow S$ proper and flat of complex spaces with $n$ pure dimensional fibers (or locally noetherian, excellent schemes with n-equidimensional fibers) the two functor $G\rightarrow f^{*}G\otimes \omega^{n}_{X/S}$ and $G\rightarrow H^{-n}(f^{!}G)$ agree?

If $f$ is Cohen Macaulay morphism, it is obviously true. But i dont know if that only case for which the two functor 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, another than CM-morphism, which satisfies this assumptions?

2) Can we relax the hypothesis on the fibers?

Thank you very much.