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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, \begin{equation} \label{2} \psi: f^{*}\mathcal{A} \longrightarrow \bigoplus_{d\geq 0}\mathcal{L}^{\otimes d} \end{equation} 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?

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    $\begingroup$ Don't take global sections, just take $B$ to be a single point. Then $f_*=\Gamma( T, \ \ )$. $\endgroup$ – Sándor Kovács Aug 2 at 6:26
  • $\begingroup$ @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. $\endgroup$ – Luke Aug 2 at 6:29

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