Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.

I want to test when $\varphi$ is the zero map. What I have is the following information:

For each point $y \in Y$ the induced map

$\varphi_{|X_y}:\mathcal U_{|X_y} \rightarrow \mathcal V_{|X_y}$

is zero. Here, $X_y$ denotes the scheme-theoretic fiber of $y$, and the notation $(.)_{|X_y}$ means that one takes pullback along the natural map $X_y\rightarrow X$.

Is this information sufficient to conclude that $\varphi =0$ ? If not, are there additional assumptions on the schemes or on $f$ that would imply the vanishing of $\varphi$ ?

Note that I do not want to assume that the cokernel of $\varphi$ is locally free, and I also would like to avoid the assumption that $X$ is integral.

Maybe to consider the special case $Y=X$ and $f=id$ can also be enlightening.

Any helpful comment on this would be highly appreciated.

Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.

I want to test when $\varphi$ is the zero map. What I have is the following information:

For each point $y \in Y$ the induced map

$\varphi_{|X_y}:\mathcal U_{|X_y} \rightarrow \mathcal V_{|X_y}$

is zero. Here, $X_y$ denotes the scheme-theoretic fiber of $y$, and the notation $(.)_{|X_y}$ means that one takes pullback along the natural map $X_y\rightarrow X$.

Is this information sufficient to conclude that $\varphi =0$ ? If not, are there additional assumptions on the schemes or on $f$ that would imply the vanishing of $\varphi$ ?

Note that I do not want to assume that the cokernel of $\varphi$ is locally free. If necessary we may assume $X$ as reduced.

Maybe to consider the special case $Y=X$ and $f=id$ can also be enlightening.

Any helpful comment on this would be highly appreciated.

Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.

I want to test when $\varphi$ is the zero map. What I have is the following information:

For each point $y \in Y$ the induced map

$\varphi_{|X_y}:\mathcal U_{|X_y} \rightarrow \mathcal V_{|X_y}$

is zero. Here, $X_y$ denotes the scheme-theoretic fiber of $y$, and the notation $(.)_{|X_y}$ means that one takes pullback along the natural map $X_y\rightarrow X$.

Is this information sufficient to conclude that $\varphi =0$ ? If not, are there additional assumptions on the schemes or on $f$ that would imply the vanishing of $\varphi$ ?

Note that I do not want to assume that the cokernel of $\varphi$ is locally free, and I also would like to avoid the assumption that $X$ is integral.

Any helpful comment on this would be highly appreciated.

Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.

I want to test when $\varphi$ is the zero map. What I have is the following information:

For each point $y \in Y$ the induced map

$\varphi_{|X_y}:\mathcal U_{|X_y} \rightarrow \mathcal V_{|X_y}$

is zero. Here, $X_y$ denotes the scheme-theoretic fiber of $y$, and the notation $(.)_{|X_y}$ means that one takes pullback along the natural map $X_y\rightarrow X$.

Is this information sufficient to conclude that $\varphi =0$ ? If not, are there additional assumptions on the schemes or on $f$ that would imply the vanishing of $\varphi$ ?

Note that I do not want to assume that the cokernel of $\varphi$ is locally free, and I also would like to avoid the assumption that $X$ is integral.

Maybe to consider the special case $Y=X$ and $f=id$ can also be enlightening.

Any helpful comment on this would be highly appreciated.