# Representability of Grassmannian functor by a scheme

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.

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.
• Thank you for this answer, that does help a lot. I was wondering if I could clarify what you said about being sufficient to check this on affines. I know that once you show the functor is a Zariski sheaf, then you can assume WLOG that the base scheme $S$ is affine. But is reducing to the case where the scheme $T \rightarrow S$ is affine in EGA? Or is this what I have heard was developed by Deligne using projective limits of affines? – Luke Jun 18 '19 at 6:06