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Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\mathcal{O}_{X \times Y}$-module, say $G$ on $X \times Y$ flat over $Y$ containing $F$ such that for each $y \in Y$,

1) $G|_{X_y}$ is injective and

2) $F|_{X_y} \subset G|_{X_y}$, where $X_y:=X \times \{y\}$?

Any reference/hint will be most welcome.

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    $\begingroup$ You certainly cannot arrange the last two conditions that $I|_{X_y}$ is injective and $F|_{X_y}\to I|_{X_y}$ is injective. Consider the case that $X$ equals $\text{Spec}\ k = \text{Proj}\ k[t]$ and $Y$ is $\mathbb{P}^1 = \text{Proj}\ k[u,v]$. Let $F$ be the structure sheaf. Since every injective sheaf is divisible, $I|_{X_y}$ is a zero sheaf. Yet $F|_{X_y}$ is the structure sheaf of $X_y$. $\endgroup$
    – Jason Starr
    Commented Feb 6, 2017 at 4:51
  • $\begingroup$ @JasonStarr I have modified the question. I no longer assume that the sheaf over $X \times Y$ is injective. Is this now possible or is it false as well? $\endgroup$
    – user45397
    Commented Feb 6, 2017 at 12:35
  • $\begingroup$ If $G$ is coherent, then $G|_{X_y}$ is also coherent. For $X_y$ equal to $\mathbb{P}^1_{\kappa(y)}$, for instance, the only coherent sheaf that is also injective is the zero sheaf. $\endgroup$
    – Jason Starr
    Commented Feb 6, 2017 at 13:00
  • $\begingroup$ @JasonStarr Thank you. One last question. Is the question still wrong if instead of coherent, I ask for quasi-coherent (at all places)? $\endgroup$
    – user45397
    Commented Feb 6, 2017 at 13:27
  • $\begingroup$ "... I ask for quasi-coherent (at all places)?" You could take a direct sum over all points $y$ of $Y$ of the pushforward from $X_y$ of an injective quasi-coherent sheaf that contains $F|_{X_y}$. $\endgroup$
    – Jason Starr
    Commented Feb 6, 2017 at 13:35

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