Representability of Grassmannian functor by a scheme

Question

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.

Answer 1

First of all, being quasicompact is not a "strong finiteness assumption", come on :). For what you're doing you're actually free to restrict $F$ to quasi-compact quasi-separated schemes, or even just affine schemes over $S$, because the inclusion of sites $Aff/S \to Sch/S$ induces an equivalence of categories of sheaves (because every scheme is locally affine).

You don't need that here though. I think you misunderstood Grothendieck's argument. For any $\Gamma(S, O_S)$-module $E$, you can find a surjection $t$ onto $E$ from some $O_S^{\oplus I}$ with $I$ possibly an infinite set. For example, you can just take $I$ to be the set of elements in $E$ and take the sum of the maps $O_S \to E$ determined by every $x \in E$. Whatever your choice of $t$, for every element $i \in I$, take the corresponding map $t_i : O_S \to E$ and let $t_{i,T} : O_T \to g^*E$ be its restriction to $T$.

Now here's the family of subsets. Let $F_i(T) \subset Grass_1(g^*E)$ be the set of line bundle quotients $g^*E \twoheadrightarrow H$ such that the induced map $$ O_T \xrightarrow{t_{i,T}} g^*E \twoheadrightarrow H $$ is surjective (hence bijective). As Grothendieck notes, this clearly defines a sub functor as $T$ varies over $S$-schemes.