I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, let $\mathcal{E}$ be a quasicoherent sheaf on a scheme $S$ and we define the functor $F: Sch/S \rightarrow \text{Set}$ by (just thinking about the n=1 case since that should clarify all I need): $$ F(T \stackrel{g}{\rightarrow}S) = \{ \text{invertible sheaves } \mathcal{L} : (g^{*}\mathcal{E} \rightarrow \mathcal{L} \rightarrow 0 )/\sim \} $$ where the equivalence $\sim$ is just isomorphisms commuting with the quotient in the obvious way.

It is easy enough to see that $F$ is a Zariski sheaf. So consider when $S = \text{Spec} A$ is affine. All I need to do is cover $F$ by representable open subfunctors, which is where I run into a problem.

From what I understand, Grothendieck's argument is as follows: Let $\mathcal{E} = \tilde{M}$ be generated by sections $\{ m_{i} \}_{i \in I}$. Then the sections $\{ g^{*}m_{i} \}_{i \in I}$ generated $g^{*} \mathcal{E}$ on $T$ and so correspond to a surjection, $$ \bigoplus_{i \in I} \mathcal{O}_{T}^{(i)} \longrightarrow g^{*} \mathcal{E} \longrightarrow 0. $$ He then seems to appeal to the fact that for the $n=1$ case, such a surjection must factor through precisely one of the summands. This is where I get lost. It looks like some kind of compact object argument, but any argument I can see would rely on strong finiteness assumptions on the scheme $T$, such as $T$ being at least quasicompact (and probably quasiseparated). Can anyone explain how the subfunctors are defined, and how they go on to cover $F$?

For reference, I found this note which seems to suggest quasicompactness is necessary also.


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