# Fiberwise torsion free and generically null sheaf for flat morphism

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.

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This question has some ambiguous points that need to be clarified. Is $A$ required to be coherent? What is the difference between saying $A$ is "canonically null" (or "trivially null") and saying $A = 0$? Is $F$ required to be an analytic set? In (2), do you mean to require $F \cap X_s$ is nowhere dense in $X_s$ for 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? – BCnrd Jul 28 '10 at 11:59

Dear Brian,

A is a coherent sheaf on $X$ which is null on dense open subset containing the smooth locus of $f$ and $F\cap X_{s}$ is nowhere dense in $X_{s}$.
for $f:X\rightarrow S$ flat morphism of reduced complex spaces with purely $n$-dimensional fibers and $S$-flat relative canonical sheaf $\omega^{n}_{X/S}$, is the canonical morphism $$\Theta:f{*}G\otimes \omega^{n}_{X/S}\rightarrow {\cal H}^{-n}(f^{!}G)$$
injectiv (or $S$-injective) for all torsion free coherent sheaf $G$ on $S$ ?
Remarks: for a local parametrization $h:X\rightarrow Y$ with $Y$ $S$-smooth of relative dimension $n$, $Theta$ is described by the morphism: $$q^{*}G\otimes {\cal H}om(h_{*}O_{X}, O_{Y})\rightarrow {\cal H}om(h_{*}O_{X}, q^{*}G)$$ where $q$ is the canonical projection of $Y$ on $S$. – kaddar Jul 28 '10 at 13:59
Dear Mohamed: if you can avoid opening a new account each time you ask a question then you'll build up reputation points and can then make comments. You can also always edit your own questions directly (and so if you edit this question then you can delete your answer and comment above). I don't know what "$S$-injective" means (something on fibers?), but for the motivating question why can't you restrict to a dense Zar. open in $S$ to reduce to the easy case when $G$ is a vector bundle? Also, why do you ask so many questions about torsion-free coherent sheaves? – BCnrd Jul 29 '10 at 10:08