# Morphisms of flat families of sheaves

$$X$$: projective scheme over a scheme $$S$$.

$$E, F$$: $$\mathscr{O}_X$$-modules, flat/$$S$$

$$\phi$$: $$E \rightarrow F$$ : morphism s.t. $$\phi_t$$: $$E_t \rightarrow F_t$$ is zero morphism for all $$t \in S$$

Then, is $$\phi$$ zero morphism ?

Edit: especially I am interested in the case $$X = Y \times S$$ ,where $$Y$$: projective surface / $$\mathbb{C}$$, $$S: \mathbb{C}$$-scheme
No. For instance, take $$X = S = \mathrm{Spec}(\Bbbk[\epsilon]/\epsilon^2), \qquad E = F = \Bbbk[\epsilon]/\epsilon^2, \qquad \phi = \epsilon.$$ Then for the unique point $$t \in S$$ the morphism $$\phi_t$$ is zero, while $$\phi$$ is not.
• Thank you for the answer! If$Y, S$: $\mathbb{C}$-schemes and $Y$: projective over $\mathbb{C}$ of dim $>0$ and $X := Y \times S$,then is the question true ? – Walter field Dec 24 '19 at 15:51
• @Walter field: I think you can cross the above example by P1. Let everything be over $\mathbb C$, $D:=Spec(\mathbb{C}[\epsilon]/(\epsilon^2)$, $X:=D\times\mathbb{P}^1$, $\pi:X\to D$, $E:=I$ where $I=\pi^{*}(\epsilon)$, $F:=\mathcal O_X$, and $E\to F$ the inclusion. – Qfwfq Dec 24 '19 at 16:25
• Thank you for the comment! But, how do we prove $E$ is flat over $D$ ? – Walter field Dec 25 '19 at 2:00