# Direct sums of invertible sheaves commuting with global sections and the functor of points approach

I am looking at the Stacks Project's treatment of the functor of points for projective space.

Let's restrict to the case that $$S$$ is a graded ring, generated by $$S_{1}$$ as an $$S_{0}$$ algebra. The scheme $$\text{Proj} S$$ represents the functor which assigns to a scheme $$Y$$ the set of (equivalence classes of) pairs $$(\mathcal{L}, \psi)$$, where $$\mathcal{L}$$ is an invertible sheaf on $$Y$$ and $$\psi$$ is a morphism of graded rings, $$\psi: S \longrightarrow \bigoplus_{d \geq 0} \Gamma(Y, \mathcal{L}^{\otimes d}) \qquad (*)$$ such that the image of $$S_{1}$$ under $$\psi_{1}$$ generates $$\mathcal{L}$$ as global sections.

I understand this, but I wanted to see this as the special case of the more general relative projective space of a quasicoherent sheaf of graded algebras.

Let the base scheme $$B=\text{Spec}R$$ be affine noetherian. Let $$\mathcal{A}$$ be a quasicoherent sheaf of graded algebras on $$\text{Spec}R$$ generated by $$\mathcal{A}_{1}$$. The scheme $$\underline{Proj} \mathcal{A}$$ represents the functor that assigns to a $$B$$-scheme $$f:T\rightarrow B$$ the equivalence class of pairs $$(\mathcal{L}, \psi)$$ with $$\mathcal{L}$$ an invertible sheaf on $$T$$ and now $$\psi$$ a morphism of graded sheaves of algebras, $$$$\label{2} \psi: f^{*}\mathcal{A} \longrightarrow \bigoplus_{d\geq 0}\mathcal{L}^{\otimes d}$$$$ which is surjective in degree 1.

I can't see how the first is a special case of the second, which it should be.

Indeed, take $$\mathcal{A}$$ to be $$\widetilde{S}$$, then the data of $$(\mathcal{L}, \psi)$$ in the relative case gives us $$\psi: f^{*}\widetilde{S} \longrightarrow \bigoplus_{d\geq 0}\mathcal{L}^{\otimes d}.$$ Using the adjunction $$f^* \dashv f_{*}$$ this is equivalent to $$\psi^{\mathfrak{a}}: \widetilde{S} \longrightarrow f_{*}\bigoplus_{d\geq 0}\mathcal{L}^{\otimes d}.$$ Since this is now a morphism of sheaves on an affine scheme, we can use the sheafification - global section adjunction for quasicoherent sheaves to say this is equivalent to a mormorphism of rings, $$S \longrightarrow \Gamma \left(T, \bigoplus_{d\geq 0}\mathcal{L}^{\otimes d} \right).$$ This looks frustratingly similar to equation (*), but unless $$T$$ is quasi-compact and quasi-separated, the direct sum won't commute with the global sections. So the data is not equivalent.

What am I missing here?

• Don't take global sections, just take $B$ to be a single point. Then $f_*=\Gamma( T, \ \ )$. – Sándor Kovács Aug 2 at 6:26
• @SándorKovács sorry maybe I'm not understanding, but how does that help? That would solve it in the case where B is a one point space. – Luke Aug 2 at 6:29