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I have some questions about flat families of sheaves. Would someone mind telling me about the following questions ?

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 $Hom(E,F)=0$ ?

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 $Hom(E,F)=0$ ?

I have some questions about flat families of sheaves. Would someone mind telling me about the following questions ?

Let $S$ be a noetherian 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: 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)) , then $E$ coherent ?

Question2: In addition, if $F$ 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 $Hom(E,F)=0$ ?

I have some questions about flat families of sheaves. Would someone mind telling me about the following questions ?

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 $Hom(E,F)=0$ ?

I have some questions about flat families of sheaves. Would someone mind telling me about the following questions ?

Let $S$ be a noetherian scheme over algebraically closed field $k$, $X$ be a projective scheme over $k$, and $E$ be a quasi-coherent sheaf on $X \times_k S$.

Question1: 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)) , then $E$ coherent ?

Question2: In addition, if $F$ 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 $Hom(E,F)=0$ ?

I have some questions about flat families of sheaves. Would someone mind telling me about the following questions ?

Let $S$ be a noetherian 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: 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)) , then $E$ coherent ?

Question2: In addition, if $F$ 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 $Hom(E,F)=0$ ?