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Let $f:X \to Y$ be a flat, projective morphism between projective varieties. Let $F, G$ be coherent sheaves on $X$, flat over $Y$. Let $\phi_1, \phi_2$ be two morphisms from $F$ to $G$ such that:

1) the image of $\phi_1$ and $\phi_2$ are flat over $Y$,

2) for every $y \in Y$, $\mathrm{Im}(\phi_1)|_{X_y} \subset G|_{X_y}$ and $\mathrm{Im}(\phi_2)|_{X_y} \subset G|_{X_y}$, where $X_y:=f^{-1}(y)$.

Is it then true that the morphism $\phi_1+\phi_2:F \to G$ has the same properties i.e.,

1) the image of $\phi_1+\phi_2$ is flat over $Y$ and

2) for every $y \in Y$, $\mathrm{Im}(\phi_1+\phi_2)|_{X_y} \subset G|_{X_y}$?

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  • $\begingroup$ That is not true. For $X=Y=\text{Proj}\ k[u,v,w]$, for $F=\mathcal{O}_X^{\oplus 3}$, for $G=\mathcal{O}_X(1)$, for $\phi_1=(u,v,w)$, and for $\phi_2 = (-u,u-v,v-w)$, both $\phi_1$ and $\phi_2$ are surjective, but $\phi_1+\phi_2$ has image $\mathcal{I}\cdot G$, where $\mathcal{I}$ is the ideal sheaf of the closed point of the homogeneous prime ideal $\langle u,v\rangle$. $\endgroup$
    – Jason Starr
    Commented Feb 7, 2017 at 0:55

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This shouldn't be the case.

Let $X = Y = \mathbb{P}^1$ with projective coordinates $[x:y]$, $f = id_X$. $F = \mathcal{O}_X$, $G = \mathcal{O}_X \oplus \mathcal{O}_X(1)$. Let $\phi_1 = id \oplus x$ and $\phi_2 = (-id)\oplus x$.

Then $\phi_1 + \phi_2 = 0\oplus 2x$, which has cokernel $\mathcal{O}_X \oplus \mathcal{O}_{[0:1]}$ and $Im(\phi_1 + \phi_2)\cong \mathcal{O}_X$ (as the map $\phi_1 + \phi_2$ is an injective map of sheaves).

So, at the point $[0:1]$ we have

$Im(\phi_1+\phi_2)|_{[0:1]} = \mathcal{O}_X|_{[0:1]} = \mathbb{C}\xrightarrow{(\phi_1+\phi_2)|_{[0:1]} = 0\oplus 0} \mathbb{C}\oplus\mathbb{C} = G|_{[0:1]},$

which violates your condition (2).

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