# About flat families of sheaves

I have some questions about flat families of sheaves.

Let $$S$$ be a scheme over algebraically closed field $$k$$, $$X$$ be a projective scheme over $$k$$, and $$E,F$$ be quasi-coherent sheaves of finite presentation on $$X \times_k S$$.

Question1:$$E,F$$ coherent ?

Question2: In addition, if $$E$$ is a flat family of torsion free coherent sheaves on $$X$$(i.e.E is flat over $$S$$ and $$E_t$$ is torsion free coherent on $$X (\forall t \in S$$ : closed point)) ,$$F$$ is also a flat family of torsion free coherent sheaves on $$X$$, and $$Hom(E_t,F_t)= 0(\forall t \in S$$ : closed point),then $$Hom(E,F)=0$$ ?