Hi.

Has some one an example of sheaf $A$ on flat morphism $f:X\rightarrow S$ of reduced complex spaces with fibers of constant positive dimension (or locally noetherian excellent schemes without embedded components) which satisfies the properties :

1) there is a dense open subset $V$ of $X$ (smooth or Cohen-Macaulay locus) on which $A$ is canonically null,

2) For every subset $F$ s.t $F\cap f^{-1}(s)$ has empty interior in $f^{-1}(s)$, we have ${\cal H}^{0}_{F}(A) = 0$,

3) There is some fibers on which the restriction of $A$ is not trivialy null.

Thank you.

all$s \in S$? In (1), do you mean to say "for example" inside of the parentheses? I wonder why the following isn't an example: take $X$ to be the affine line over $S$ and $A$ the structure sheaf of a single fiber. Lastly, can you give some motivation for the question? $\endgroup$ – BCnrd Jul 28 '10 at 11:59