Quot scheme on open subset

Question

Let $X$ be a scheme, and $U$ an open subset of $X$. If $X$ is projective and $F$ some coherent sheaf on it, then the Quot scheme $Quot_{X}(F)$ exists, and $Quot_{U}(F|_{U})$ too. How can one describe the embedding of $Quot_{U}(F|_{U})$ into $Quot_{X}(F)$. I.e., given a quotient $F|_{U}\rightarrow Q$, how do we get a quotient of $F$? I think we should take the composite $F\rightarrow i_{\ast} i^{\ast} F\rightarrow i_{\ast} Q$, but why should this be surjective? Since $i:U\rightarrow X$ is an open immersion, the pushforward is not exact.

Answer 1

Your intuition is correct. The map $p\colon F \to i_\ast i^\ast F \to i_\ast Q$ is still surjective. In short, this is because you can check this after replacing $i\colon U\hookrightarrow X$ with $i|_{\textrm{Supp}(Q)}\colon \textrm{Supp}(Q) \,\,\widetilde{\to}\,\, \textrm{Supp}(i_\ast Q)$.

First of all, $\alpha\colon i^\ast i_\ast Q \,\,\widetilde{\to}\,\, Q$ is an isomorphism because $i$ is an immersion. The surjection $F|_U \twoheadrightarrow Q$ you started with is recovered as $\alpha\circ p|_U \colon F|_U \to i^\ast i_\ast Q \,\,\widetilde{\to}\,\, Q$, thus $p|_U \colon F|_U \to i^\ast i_\ast Q$ is surjective. But $i$ is an isomorphism around the support of $Q$ and our calculation takes place precisely there; thus we may assume $i$ is surjective, so $i^\ast$ is in particular faithfully flat, hence $i^\ast$ reflects epimorphisms, hence $p$ is surjective.

See Lemma A.1 here for a generalisation.